math problem solving examples for grade 2

Examples of Second Grade Word Problems with Solutions

In today’s post, we are going to see a selection of Second Grade word problems that usually appear at the end of a Smartick session for children with a second-grade level . You will also get a chance to see the solution for each problem.

Ready? Let’s go!

How To Correctly Solve Any Problem

• Read the word problem carefully! You should understand what is being asked very clearly.
• Write down the data.
• Complete the operations and always write down the results for each.
• Write the solution very clearly and respond to what the problem asks.

Second Grade Word Problems

Let’s practice with a selection of Second Grade word problems from Smartick.

Word Problem 1

For this word problem, Smartick has given us the solution and it is up to us to select the correct question.

The answer is: ”They were planting 38 chestnut seeds.” Let’s take a look at each option to see which would work with this answer, we’ll use the process of elimination.

• How many more chestnut seeds than oak seeds were they planting?  This is asking us about the different types of seeds, but the solution is only asking about chestnuts so this question is not what we are looking for.
• Are fewer chestnut trees or fewer oak trees expected to grow? This answer isn’t the one we are looking for because the answer only gives us information about chestnut seeds.
• How many more poplar seeds than oak seeds were they planting? Nothing was mentioned in the word problem or the answer about poplar trees.
• How many chestnut seeds were they planting? Here were are being asked about the number of chestnut seeds that were planted and the solution tells us the amount that was planted. This is the question that we are looking for.

Word Problem 2

This word problem is asking how many tablespoons of oil have been added to a stew, and we know that there were 20 tablespoons more added than the 9 tablespoons that the recipe called for. So we need to know how many tablespoons in total were added to the stew.

9 + 20 = 29

29 tablespoons of oil were added to the stew. 

Word Problem 3

Here, we are being asked to calculate the total number of clocks. To help us solve it, we can create a drawing using the information from the word problem. It tells us that there are 2 clocks on each shelf, and there are 6 shelves.

Word Problem 4

First, we are asked which operation we need to use to solve the problem and are provided with the options of addition and subtraction. Let’s read the word problem carefully to see which would work best…

There were 6 oranges before Diego added more, and now there are 10. Therefore, the difference between the amount in the bowl now, and the amount there was before, is the amount that Diego added. If we add the numbers, adding the larger quantity to the lesser quantity, it does not give us the number of oranges Diego added to the bowl. However, if we subtract the smaller quantity from the larger quantity, we should find the difference.

The answer to the first question would be: 10 – 6.

And the solution to the word problem: 10 – 6 = 4 .

Diego added 4 oranges. 

I hope you have learned something new from this selection of second grade word problems that appear during our daily Smartick sessions. If you would like to learn more about other primary school math topics, register with Smartick and try it for free.

Learn More:

• What Do Double and Half Mean?
• Examples of 3rd Grade Math Word Problems with Solutions
• How to Solve Multiplication Word Problems
• Alternative Word Problems at Smartick
• Singapore Bars Applied to Fractions
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Second Grade Math Problems

Welcome to the Second Grade Math Problems page. We have a wide selection of longer math problems requiring a wide range of math skills to solve.

These problems are also a great way of developing perseverance and getting children to try different approaches in their math.

On this webpage are our selection of longer, more in-depth problem solving sheets for 2nd grade.

Typically, there is just one problem on each page with maybe a follow up problem in some cases.

The sheets cover a wide range of Math topics, from place value and number fact knowledge to geometry and logic problems.

The following worksheets have been designed to develop a wide range of skills and problem solving techniques such as:

• making lists or tables
• drawing pictures to help solve problems
• working systematically
• logical thinking
• number fact knowledge
• persevering until all solutions have been found

An answer sheet is available for each worksheet provided, where appropriate.

These sheets can be used in many different ways:

• to challenge more able pupils
• to use as a way of developing strategies to explore more in-depth problems, such as making lists or tables
• to use as an extension activity for children who finish early
• to use as part of a Maths challenge board
• Broken Calculator Problem 1

The Broken Calculator problem is a number problem involving using an imaginary broken calculator with only the 2, 3, + and = buttons working to make different totals.

There are 2 versions of the problem sheet, one with a pre-prepared template for filling in, and a second blank version for children to show their own recording system.

• No table version
• PDF version

Anyone for an Ice Cream?

Anyone for an Ice Cream is a money activity which involves using silver coins only to make a total of 40¢ . The aim is to find all the possibilities.

• Anyone for an Ice-cream?
• Anyone for an Ice Cream? UK version
• Tyger's Coin Challenges

Tyger's Coin Challenge is a money activity. The aim is to see whether or not different amounts of money can be made from a number of coins.

• Captain Salamander's Letter

This 2nd grade math problem sheet involves working out which totals of money can be made using only 3¢ and 5¢ stamps. It is a good activity for developing perseverance and logical thinking.

• Balloon Pairs #2

Balloon Pairs is a number adding activity where the aim is to find different totals by adding the balloon numbers together. The totals are then sorted in order of size using a table.

• Balls in the Bucket Challenge #2

This challenge involves working out how different scores were made in the balls-in-the-bucket game. It is a 'finding all possibilities' type of problem.

• Birthday Girl

Birthday Girl is an activity which involves finding the correct ages of all the people in the challeges using the clues that are given.

• Climb the Mountain

This is one of our second grade math problems that involves finding all the possible paths up to the top of the mountain using the routes provided.

• Dilly's Eggs #1

Dilly's eggs is a sharing problem - drawing it out is a good strategy for tackling this problem. The aim is to find the number of eggs Dilly had using the clues provided.

• Odd Square Out

This is a good activity for developing noticing skills and recognising shapes that have been rotated or reflected.

• Parking Lots #2

Parking Lots is an activity where the aim is to find as many combinations as possible for the cars to park. Systematic working could be an area of focus for this activity.

• Pick the Cards #2

Pick the Cards is an adding game where the aim is to use combinations of numbers to reach a given total. This activity is good for adding three or four small numbers together to make a given total.

• Place It Right #2

Place It Right is a place value activity to support children with their place value learning. The aim is to make a range of 3 digit numbers with different properties.

• Share the Treasure #2

Share the Treasure is a logic acitivity where the aim is to share some treasure according to certain criteria.

• Who Chose Which Shape #2

Who Chose Which Shape is a logic problem where children have to work out which salamander chose which shape from the clues given.

Looking for some easier word problems

We have a range of easier word problems at our parent site, math-salamanders.com

The problems on this page are at a simpler level than those here.

Many of the problems, e.g. Dilly's Eggs, Pick the Cards and Share the Treasure have easier versions on this page.

Using the link below will open our main site in a new tab.

• First Grade Math Problems

Looking for some harder word problems

We have a range of more challenging word problems at our parent site, math-salamanders.com

The problems on this page are at a trickier level than those here.

Some of the problems, e.g. Place It Right and Share the Treasure have harder versions on this page.

• 3rd Grade Math Problems

Addition and Subtraction Puzzles

The puzzles in this section mainly focus on adding and subtracting numbers.

The puzzles start with adding and subtracting to 20, and progress on to harder levels and more complex puzzles.

Using the puzzles in this section will help your child to:

• develop their adding and subtracting skills;
• develop trial and improvement strategies;
• improve problem solving skills.

All the second grade math problems in this section will help your child to learn their addition and subtraction facts and become more confident with handling numbers mentally.

• Free Math Puzzles - Addition and Subtraction

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2nd Grade Word Problems

Lessons on solving word problems using tape diagrams, block diagrams or bar models, for students in Grades 2.

Related Pages Grade 2 Math Lessons Singapore Math Lessons Arithmetic Worksheets Arithmetic Lessons

The following are some examples of 2nd Grade Word Problems for addition, subtraction, multiplication and division. These word problems are solved with the help of "thinking blocks" or "block diagrams" which are used in Singapore Math or tape diagrams used in Common Core Math.

The following diagrams show how Part-Part-Whole Model, Comparison Model, Equal Parts of a Whole Model. Scroll down for examples and solutions.

Example: There are 56 passengers in Bus A and 39 passengers in Bus B. How many passengers are there altogether in the two buses?

So, 56 + 39 = 95 There are 95 passengers altogether in the two buses.

Subtraction

Example: A fruit seller had 84 lemons. He sold 56 lemons. How many lemons did he have left?

So, 84 – 56 = 28 He had 28 lemons left.

Multiplication

Example: There are 2 papayas in each box. How many are there in 4 boxes?

4 × 2 = 8 There are 8 papayas in 4 boxes.

Example: Alex bought a sack of 27 kg of flour. He packed the flour equally into 3 bags. How many kilograms of flour were there in each bag?

27 ÷ 3 = 9 There were 9 kg of flour in each bag.

Helping Parents Explain Math - Grade 2 Word Problems

How to use “thinking blocks” to help even young children - 1st grade, 2nd grade - understand, model and solve word problems.

• David has 45 marbles. His sister gave him 32 more. How many marbles does he have altogether?
• Caleb wants to buy a scooter that costs $63. He saved $38 so far. How much more money does Caleb need?

Helping Parents Explain Math - Word Problems 2

• Anna is riding in a bike-a-thon. She rode 48 miles in the morning, stopped for a lunch break, then rode another 37 miles in the afternoon. Anna was then only 62 miles from the finishing line. How long was Anna’s bike ride?
• Kierka bought a juice for 55 cents, a breakfast bar for 27 cents and a package of gum. She spent $1/22 altogether. How much did the package of gum cost?

Adding with Thinking Blocks or Block Diagrams

Example: Alicia has $6 more than Bobby. If Bobby has $10, how much do they have altogether?

Subtracting with Thinking Blocks or Block Diagrams

Example: Mr. Oliver bought 88 pencils. he sold 26 of them. How many pencils did he have left?

Multiplying with Thinking Blocks or Block Diagrams

• Ben the Baker puts three brownies into each gift box. How many brownies should Ben the Baker bake in order to fill ten gift boxes?
• Blake donated his collection of adventure books to a local school. He packed ten books in each of his three boxes. How many books did Blake donate?

Model Drawing for Second Grade

• Step-by-step visual method of turning a word problem into a diagram with unit bars which represent values.
• Can make a complicated problem seem easy.
• Provides a bridge from the pictorial to abstract. It is a tool.
• 80% of word problems can be solved with model drawing (50% in middle and high school).
• Students start to become less intimidated by math and start enjoying it.

How to solve Addition Problems using Part-whole Models?

• Sean has 2 plastic cars. He also has 5 wooden cars. How many cars does Sean have altogether?
• Amy has 3 green marbles. She also has 6 blue marbles. How many many marbles does Amy have altogether?

How to solve Addition Problems using Comparison Models?

• Jesse had $3.00 more than Clinton. If Clinton had $10.00, how much money did they have altogether?
• Penny ate 6 ice cream cones more than Jeff did over the weekend. If Jeff ate 8 ice cream cones, how many ice cream cones did they eat altogether?

How to solve subtraction problems using Part-whole Models?

• Nathan had $27 to buy gifts for his family. If he spent $9.00 on a gift for his brother, how much money did he have left to spend on the rest of the family?
• Mike has 21 crayons. Elisha has 6 crayons. how many more crayons does Mike have?
• Tom has 97 pennies. Samantha has 29 pennies. How many more pennies does Tom have?

Mixed Operation Bar Model Problem

• Josh scored 27 points in the championship basketball game. That was 12 points more than his friend Luke scored. How many points did they score altogether?

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Second Grade Math Worksheets and Printables

Calculate the Fun with Second Grade Math Worksheets

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Addition Word Problems 2nd Grade Addition Problems within 100

Welcome to our Addition Word Problems 2nd Grade Worksheets. Here you will find a wide range of free printable addition word problem worksheets, which will help your child practice solving a range of addition problems using numbers with a sum of up to 100.

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• Addition Word Problems within 100
• Addition Word Problems within 100 with 3 addends
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• Addition Word Problems up to 100 Online Quiz

Addition Word Problems 2nd Grade

Addition problems within 100.

Each sheet consists of adding two or three numbers with a total of up to 100.

There is a space on each sheet for working out, so that your child can write out the problem and solve it.

We have split the worksheets up into word problems with and without regrouping.

Using these sheets will help your child to:

• add up two or three numbers within 100;
• solve addition word problems with and without regrouping.

Addition Word Problems 2nd Grade within 100

There are two versions of each sheet.

The first version (version A) contains problems where no regrouping is needed.

The second version (version B) contains similar problems but regrouping is needed to solve them.

Sheets 1A, 1B, 2A, 2B, 3A and 3B have just two addends to add up.

Sheets 4A, 4B, 5A and 5B have three addends to add together.

Addition Word Problems within 100 with 2 Addends

• Addition Word Problems within 100 Sheet 1A (no regrouping)
• PDF version
• Addition Word Problems within 100 Sheet 1B
• Addition Word Problems within 100 Sheet 2A (no regrouping)
• Addition Word Problems within 100 Sheet 2B
• Addition Word Problems within 100 Sheet 3A (no regrouping)
• Addition Word Problems within 100 Sheet 3B

Addition Word Problems within 100 with 3 Addends

• Addition Word Problems within 100 Sheet 4A (no regrouping)
• Addition Word Problems within 100 Sheet 4B
• Addition Word Problems within 100 Sheet 5A (no regrouping)
• Addition Word Problems within 100 Sheet 5B

Looking for some easier worksheets?

Take a look at our Addition word problems for first graders.

On this page, your child will learn to work out basic addition word problems with sums up to 20.

• 1st grade Addition Word Problems

Looking for some harder worksheets?

We have a range of 3-digit addition worksheets set out in columns.

• Addition Word Problems 3rd Grade (3- and 4-digits)
• 3-Digit Column Addition Worksheets

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Addition & Subtraction Worksheets 2nd Grade

• Add and Subtract Within 20 Worksheets
• 3 Digit Addition and Subtraction Worksheets

More 2nd Grade Addition Worksheets

Here you will find some more of our 2nd Grade Addition Worksheets.

The link below will open our 2nd-grade-math-salamanders website in a new browser window.

• Addition Word Problems 2nd grade at 2nd-grade-math-salamanders.com
• Number Bonds to 20
• Math Addition Facts to 20

More 2nd Grade Math Word Problems

Here are a range of problems solving sheets for 2nd graders. Most of the sheets contain 'real-life' problems related to animal facts.

Using the sheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

These sheets involve solving one or two more challenging longer problems. This link opens in a new tab.

• Second Grade Math Problems

These sheets involve solving many 'real-life' problems involving data.

• 2nd Grade Math Word Problems

These sheets involve solving a range of addition and subtraction word problems up to 100.

• 2nd Grade Addition and Subtraction Word Problems

These sheets involve solving a range of subtraction word problems up to 100.

• Subtraction Word Problems 2nd grade

These sheets involve solving a range of multiplication problems.

• Multiplication Word Problems 2nd Grade

Addition Word Problems to 100 Online Quiz

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This quick quiz tests your knowledge and skill at solving addition word problems within 100.

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50 Problems for 2 nd Graders

Practice your math skills with these 50 math problems for 2nd graders , covering addition, subtraction, place value, measurement, time, money, and geometry! Answers included.

Author Michelle Griczika

Published November 3, 2023

50 Math Problems for 2 nd Graders

Published Nov 3, 2023

Key takeaways

• Mastering addition and subtraction skills helps us solve mathematical puzzles and apply them to real-life situations.
• Understanding place value enables us to correctly read, write, and compare numbers, developing our number sense and mathematical reasoning.
• Solving measurement word problems helps us apply mathematical concepts to practical scenarios, improving our problem-solving and critical thinking abilities.
• Exploring time and money concepts allows us to understand the world around us, manage our daily routines effectively, and develop essential life skills.
• Recognizing shapes and their properties opens our eyes to the diverse world of geometry, helping us understand how objects are structured and enhancing our spatial awareness.

Table of contents

• Key Takeaways
• Addition and Subtraction
• Place Value
• Measurement Word Problems
• Time and Money

Hello, second grade mathematicians! Are you ready for some mind-boggling math challenges?

We will be going over important math skills for second graders to strengthen their understanding and knowledge.

• In second grade math , addition and subtraction skills are expanded to solve practical second grade math problems .
• Understanding place value allows us to navigate through large numbers and count objects, strengthening our number sense.
• Measurement math word problems for 2nd graders provide exciting puzzles that apply math skills to real-life scenarios, fostering problem-solving abilities and transforming us into math superstars.
• Counting money and telling time teach us essential skills for day to day life.
• Shapes are the building blocks of our surroundings. Understanding squares, triangles, and circles enables us to analyze their attributes and how they fit together.

All of these skills are necessary for 2nd grade math practice !

Unlock unlimited math questions

Put your skills to the test with fun exercises + maths games that are proven to boost ability!

Section 1: Addition and Subtraction

Section 2: Place Value

Use the greater than symbol >, less than symbol <, or equal to symbol = to answer.

Answer: <

Answer: >

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• Kindergarten

Measurement and data

Sample questions, section 3: measurement word problems.

Answer: 53 inches

Answer: 38 inches

Answer: 32 centimeters

Answer: 35 inches

Answer: 55 centimeters

Answer: 68 centimeters

Answer: 84 centimeters

Answer: 40 inches

Answer: 50 inches

Section 4: Time & Money

Look at the analog clock and write the digital time to the nearest five minutes:

What time is shown on the digital clock: 9:25 a.m. or 9:25 p.m.?

What time is shown on the digital clock: 1:50 a.m. or 1:50 p.m.?

Answer: $1.05

Answer: $5.08

Answer: 25 cents

Answer: 70 cents

Answer: $4.37

Section 5: Geometry

Answer: Triangle

Answer: Square

Answer: Pentagon

Answer: Hexagon

Answer: Equilateral Triangle

Answer: Trapezoid

Answer: Cube

Math practice for 2nd graders is super important because we use it every single day! For extra math practice that comes in the form of fun math games and interactive practice problems, try Doodle Learning’s math help app .

Lesson credits

Michelle Griczika

Michelle Griczika is a seasoned educator and experienced freelance writer. Her years teaching first and fifth grades coupled with her double certification in elementary and early childhood education lend depth to her understanding of diverse learning stages. Michelle enjoys running in her free time and undertaking home projects.

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

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20 Word Problems For 2nd Grade: Develop Their Problem Solving Skills Across Single and Mixed Topics

Emma Johnson

Word problems for second grade are an important tool for improving number fluency. The key focus of math in second grade is on ensuring students are becoming more fluent with number facts and the concept of place value. Children are starting to develop more efficient written methods by this stage and are beginning to carry out calculations with increasingly larger whole numbers.

As children progress through school, they are exposed to a wider variety of problem solving questions covering a range of concepts. In second grade, these include addition, subtraction, measurement and data.  

It is important that children are regularly exposed to reasoning and problem solving questions, alongside the fluency work each lesson. It is also important to remember that all children need exposure to reasoning and problem solving questions, not just the higher attaining students who finish quickest.

We have put together a collection of 20 word problems, aimed at second grade students.

Word Problems Grade 2 Addition and Subtraction

11 grade 2 addition and subtraction questions to develop reasoning and problem solving skills.

Place value

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In second grade, students focus on one-step problems, covering a range of topics. At this stage the majority of word problems students are tackling will have one-step, but they may also start to be introduced to simple two-step word problems. Here is a breakdown of topics that will be covered and expectations in second grade. 

Solve number problems and practical problems involving recognizing the place value of each digit of a 3-digit number; comparing and ordering numbers up to 1,000 and identifying, representing and estimating numbers using different representations.

Solve problems, including missing number problems, using number facts, place value and more complex addition and subtraction word problems .

Solve problems involving length; adding and subtracting within money word problems involving dollar bills, quarters, dimes, nickels and pennies.

Solve-step and two-step questions (For example, ‘How many more?’ and ‘How many fewer’?) using information presented in scaled bar charts, pictograms and tables.

By second grade, children are starting to learn how to use some of the formal written methods of addition and subtraction. It is important that the link between math in school and math in real-life continues to be made. Word problems are a key element in helping students to make this link. 

How to teach problem solving in second grade

When teaching math problems to second grade, it’s important to think of ways to make them fun, engaging and something the children are able to relate to. This might include acting out the problem, using concrete resources and providing visual images, to bring the problems to life. 

Children should have plenty of opportunity to talk in pairs, groups and as a whole class, to share their understanding of what is being asked and their math strategies for problem solving . The use of manipulatives is important and all children should have access to a range of math resources when solving problems like this.

Students need to be encouraged to read word problems carefully and to make sure they understand what is being asked, before attempting to tackle the problem. This is where the use of a partner and group discussion can really help children’s understanding. Students then need to think about what they already know and how they can use this to help them answer the question. Where appropriate, students should also be encouraged to draw diagrams and pictures to help them solve the question.

Here is an example:

Mason has 24 glass jars to put flowers in.

He gives 5 to Marcy and drops 2 while carrying them inside the shop.

How many glass jars does Mason have left?

How to solve:

What do you already know?

• Mason has a total of 24 glass jars.
• We know he gives Marcy 5 jars, which means we will need to subtract 5 from 4. 
• He also dropped 2, also meaning we will need to subtract.
• In second grade, children should be building confidence with adding and subtracting within 20, and should be able to do these calculations in their heads.
• Children who aren’t able to recall quickly could use counters to represent the jars, or draw a bar model to help solve it.

How can this be drawn/represented visually?

We can draw a bar model or counters to represent this problem:

• To calculate how many jars are left, we can either use or draw 24 counters. We can then remove or cross out the 5 jars given to Marcy, and then 2 jars that were broken.
• Using the bar model, we can first subtract 5 from 24, representing the jars given to Marcy, leaving 19 jars. Then subtract 2 more from 19, to represent the two jars broken.
• Mason had 17 glass jars left. 

Addition word problems for second grade

In second grade, students are exposed to a range of addition word problems, including problems involving mental addition and addition of up to 3-digits using formal written methods.

See also: Mental math second grade

A family driving on holiday travel 146 miles from home to the first service station.

They then drive a further 175 miles to reach their destination.

How far have they traveled altogether?

Answer : 321 miles

146 + 175 = 321

Elvie is buying a can of soda from a vending machine. She has put in 2 quarters, 2 dimes, and 3 nickels. 

How much is the can of soda?

Answer : 85¢

50¢ + 20¢ + 15¢ = 85¢

Jamie scored 443 on his new online game.

Jared scored 468.

How many points did they score between them?

Answer : 911 points

443 + 468 = 911

At Third Space Learning we often tie word problems into our one-to-one online tuition. With each programme personalized to the needs of each individual student, children are able to develop their problem solving skills, math fluency and grow confidence in math.

Subtraction word problems for second grade

Subtraction word problems in second grade also need to comprise of a combination of mental calculation questions and those involving formal written subtraction up to 3-digits. Children should also be starting to estimate answers and check their calculations by using the inverse.

Ahmed collects 374 stickers.

He needs 526 stickers to fill his sticker album.

How many more stickers does he need to collect?

Answer : 152 stickers

526 – 374 = 152

 A bag of carrots weigh 360g

A bag of tomatoes weighs 235g.

How much heavier is the bag of carrots?

Answer : 125g

360 – 235 = 125

 Ahmed buys a bag of candy with 200 pieces in it.

Over 2 weeks, he eats 145 pieces. How many pieces of candy does Ahmed have left?

Answer : 55 pieces of candy

200 – 145 or count up from 145 to 200.

Multi-step word problems in second grade

When children first move into elementary school, word problems are predominantly one-step. As they become more confident they can be exposed to more word problems, requiring a second step or multi-step word problems. When first introducing two-step problems, keep the numbers used in the problems low and manageable to allow students to focus on reasoning over calculations.

Oliver had 3 bags of candies.

Each bag contained 15 candies.

Oliver’s little brother ate 17 pieces of the candy. How many pieces of candy does Oliver have left?

Answer : 9 sweets

15 + 15 + 15 = 45 

45  – 17 = 28

A teacher photocopies 95 math worksheets and 80 English worksheets in one week.

Teachers can print a maximum of 300 worksheets per week. 

How many can the teacher print for other subjects?

Answer : 125 worksheets

 95 + 80 = 175

300 – 175 = 125

A flower shop picks 19 roses and 25 daisies fresh from their garden.

A customer orders a dozen flowers for a birthday gift.

How many flowers will the flower shop have left?

Answer : 32 flowers

19 + 25 = 44 flowers (roses and daisies combined)

44 – 12 = 32 flowers

We hope that this collection of word problems for second grade becomes a useful resource in your second grade math classroom. 

For more resources, take a look at our library. Third Space Learning offers a wide array of math and word problems resources for other grades. These include worksheets , end of year assessments and a range of math games and activities for students from kindergarten through to 6th grade. 

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by former Deputy Headteacher Emma Johnson and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.

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Home » Teacher & Parents Tips » Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Mathematics is an essential subject that students cannot afford to neglect. As early as grade 2, students are introduced to the basics of mathematical concepts such as numbers, addition, subtraction, multiplication, and division. An essential aspect of mathematics is problem-solving. While many students find it challenging, it can be made easy with the right approach. This article will discuss how parents and educators can turn problem-solving woes into victories and equip their grade 2 students with the skills needed to excel in math.

Table of Contents

The Importance of Problem Solving in Grade 2 Math

Problem-solving is an essential skill that students need to master to excel in math. It involves critical thinking, analysis, and decision-making. It also helps students develop their reasoning abilities and enables them to apply logic in real-life situations. In grade 2 math, problem-solving skills are essential as they help students understand mathematical concepts better. Students who can solve problems are more confident and enjoy math more than those who struggle.

Turning Problem-Solving Woes into Victories

Many students, especially in grade 2, find problem-solving challenging. However, with the right approach, it can be turned into a victory. One way to turn problem-solving woes into victories is by breaking down the problem into smaller parts. Encourage your child to read the problem carefully and identify the key elements. Once they have identified these elements, they can begin to solve the problem step by step. This approach helps students understand the problem better and reduces anxiety.

Another way to turn problem-solving woes into victories is by practicing regularly. The more students practice, the more comfortable they become with problem-solving. Encourage your child to practice regularly and provide them with different types of problems to solve. As they solve more problems, their confidence will increase, and they will become more efficient problem-solvers.

Strategies for Effective Problem Solving in Math

There are several strategies that students can use to solve problems effectively. One of the most effective strategies is the use of visual aids. Encourage your child to draw diagrams or pictures to help them understand the problem better. This approach helps students visualize the problem and enables them to make better decisions.

Another strategy is to use real-life situations to solve problems. This approach helps students understand how math can be applied in real-life situations. For example, if you want to teach your child about fractions, you can use pizza slices to help them understand the concept better.

Lastly, encourage your child to work with a partner or in a group. Group work helps students learn from one another and can be an effective way to solve problems. It also helps students develop their social skills and enables them to work collaboratively.

Fun and Engaging Math Activities for Grade 2 Students

Learning math can be fun and engaging. There are several math activities that parents and educators can use to help grade 2 students develop their problem-solving skills. One such activity is math games. Games such as Sudoku, Math Bingo, and Math Jeopardy can be used to teach students math concepts while making learning fun.

Another activity is math puzzles. Puzzles such as crosswords, word searches, and logic puzzles can be used to develop critical thinking and problem-solving skills. Additionally, math stories can be used to teach math concepts while making learning fun. Math stories can be found in storybooks or online, and they provide an interactive way to teach math concepts.

Developing Critical Thinking Skills through Math

Mathematics is an excellent way to develop critical thinking skills. Critical thinking involves analyzing, interpreting, and evaluating information to make informed decisions. In math, critical thinking is essential as it helps students understand mathematical concepts and apply them in real-life situations.

Encourage your child to think critically when solving problems. Teach them to ask questions and to consider different solutions to a problem. Additionally, encourage them to explain their reasoning and to justify their solutions. These skills are essential in problem-solving and can be applied in other areas of their lives.

Providing Support and Encouragement for your Child

Providing support and encouragement is essential when helping your grade 2 child excel in math. Encourage your child to ask questions and to seek help when they need it. Additionally, provide them with a positive learning environment and praise their efforts and progress. Celebrate their successes and encourage them to keep learning and practicing.

Equipping Your Child for Math Success in Grade 2

Mathematics is an essential subject that requires problem-solving skills. By turning problem-solving woes into victories, providing support and encouragement, and using effective strategies and fun activities, parents and educators can help grade 2 students excel in math. With the right approach, learning math can be fun and engaging while developing critical thinking skills that are essential in all areas of life.

2nd Grade Fast Math Success Workbook

2nd Grade Fast Math Success Workbook: Math Worksheets Grade 2: Numeration, Addition, Subtraction, Telling Time and More with Answers

• Comparing Numbers Within 200
• Ordering Numbers Within 200
• Skip Counting: Count By 1s and 2s
• Place Value: Ones, Tens, and Hundreds
• Addition: 1 through 100
• Subtraction: 1 through 100
• Addition: Missing Number – 1 through 100
• Subtraction: Missing Number – 1 through 100
• Ordering Numbers – 1 through 1000
• Comparing Numbers – 1 through 1000
• Counting: Count by 4 to 6
• Addition and Subtraction: Double Digit
• Addition and Subtraction: Missing Number – Double Digit
• Addition: Triple Addend – 1 through 100
• Mixed Operations: 1 through 100
• Write the Numbers Before, After, and Between
• Addition and Subtraction: 1 through 1000
• Telling Time
• Time Passages
• Addition and Subtraction Games
• Final Review

MathBear: Math Workbook Grade 2

MathBear: Math Workbook Grade 2: 2nd Grade Math Workbook: Addition, Subtraction, Multiplication, and Numeration with Answers

Math Practice Workbook Grade 2

Math Practice Workbook Grade 2: 3051 Questions to Master Essential Math Skills (Numeration, Addition, Subtraction, Multiplication, Telling Time and More) With Answer Key

• Circle the Numbers
• Comparing Numbers
• Addition: Double Digit
• Subtraction: Double Digit
• Addition: Missing Number
• Subtraction: Missing Number
• Number Before, After and Between
• Addition: Triple Digit
• Subtraction: Triple Digit
• Basic Multiplication

MathBear: Homeschool Math Workbook Grade 2

MathBear: Homeschool Math Workbook Grade 2: 2nd Grade Homeschool Math Practice Workbook: Addition, Subtraction, Multiplication, Place Value with Answers

• Place Value
• Commutative Property
• Addition Games
• Subtraction Games

MathBear: Math Curriculum Workbook Grade 2

MathBear: Math Curriculum Workbook Grade 2: 2nd Grade Math Curriculum: Numeration, Place Value, Addition and Subtraction, Telling Time with Answers

• Ordering Numbers
• Number Before, After, or Between
• Number Lines
• Place Value: Ones, Tens, Hundreds
• Subtraction
• Addition: 3 Addend
• Introduction to Multiplication
• Measure the Rectangles
• Match the Answers

MathBear: Math Practice Grade 2

MathBear: Math Practice Grade 2: 2nd Grade Math Practice Workbook: Addition, Subtraction, Multiplication, Place Value, Telling Time, Commutative Property with Answers

MathBear: Math Skills Workbook Grade 2

MathBear: Math Skills Workbook Grade 2: 2nd Grade Math Skills Practice Workbook: Addition, Subtraction, Basic Multiplication, Place Value, Math Games, and More With Answers

All Seasons Math Tests Grade 2

All Seasons Math Tests Grade 2 (Student's Edition): 100 Math Practice Pages Grade 2: Timed Math Tests: For Classroom and Homeschool

Kids Math Book Ages 6-8

Kids Math Book Ages 6-8: Math Practice workbook Grade 1-3: Addition, Subtraction, Place Value, Telling Time

• Addition Target
• Subtraction Target
• Addition Square
• Ordering Numbers: 1 to 100
• Addition Table
• Subtraction Table
• Addition: Triple Addend
• Numbers Before, After, and Between
• What time was and will it be?

Math Workbook Grade 2: Addition and Subtraction

MathBear: Math Workbook Grade 2: Addition and Subtraction: 2nd Grade Double Digit Addition and Subtraction Workbook with Answers

• Addition: within 100
• Subtraction: within 100
• Basic Addition with Regrouping
• Basic Subtraction with Regrouping

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Grade 2 Math Samples

Non-common core, common core, teks (for texas), algebraic reasoning unit.

• Unit Description

On the Beach

Students determine how many pink shells Kim found on the beach.

Given 20 goldfish and four fish tanks, students determine if there is an odd or even number of fish in each tank.

The Algebraic Reasoning Unit involves understanding number patterns within properties of numbers and operations in order to answer questions such as:

• How can you prove this number is odd or even?  
• How can you use the relationship between addition and subtraction to solve problems in which the unknown may be any one of the terms in the equation?  
• Give an equation in which the unknown may be any one of its terms, how can you create a situation to match it? How can you prove that your situation matches the equation?  
• How can you show ___ is an even/odd number?  
• What strategies can you use to determine a number is odd or even?  
• What happens when you add two like odd or even numbers?
• Standard Description

Barnyard Buddies

Students find the total number of legs on 8 cows and 10 chickens.

Ladybugs and Spiders

Students count the number of legs on three spiders and six ladybugs.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Cars and Trucks

Students determine if there is the same total number of cars and trucks on the shelves.

Students determine how many nuts Sam the squirrel hides under a rock.

• Give an equation in which the unknown may be any one of its terms, how can you create a situation to match it? How can you prove that your situation matches the equation?

TEKS standards covered: 2.7A, 2.7C

• Math Forum/Help
• Problem Solver
• College Math
• Word Problems

Math Word Problems and Solutions - Distance, Speed, Time

Problem 1 A salesman sold twice as much pears in the afternoon than in the morning. If he sold 360 kilograms of pears that day, how many kilograms did he sell in the morning and how many in the afternoon? Click to see solution Solution: Let $x$ be the number of kilograms he sold in the morning.Then in the afternoon he sold $2x$ kilograms. So, the total is $x + 2x = 3x$. This must be equal to 360. $3x = 360$ $x = \frac{360}{3}$ $x = 120$ Therefore, the salesman sold 120 kg in the morning and $2\cdot 120 = 240$ kg in the afternoon.

Problem 2 Mary, Peter, and Lucy were picking chestnuts. Mary picked twice as much chestnuts than Peter. Lucy picked 2 kg more than Peter. Together the three of them picked 26 kg of chestnuts. How many kilograms did each of them pick? Click to see solution Solution: Let $x$ be the amount Peter picked. Then Mary and Lucy picked $2x$ and $x+2$, respectively. So $x+2x+x+2=26$ $4x=24$ $x=6$ Therefore, Peter, Mary, and Lucy picked 6, 12, and 8 kg, respectively.

Problem 3 Sophia finished $\frac{2}{3}$ of a book. She calculated that she finished 90 more pages than she has yet to read. How long is her book? Click to see solution Solution: Let $x$ be the total number of pages in the book, then she finished $\frac{2}{3}\cdot x$ pages. Then she has $x-\frac{2}{3}\cdot x=\frac{1}{3}\cdot x$ pages left. $\frac{2}{3}\cdot x-\frac{1}{3}\cdot x=90$ $\frac{1}{3}\cdot x=90$ $x=270$ So the book is 270 pages long.

Problem 4 A farming field can be ploughed by 6 tractors in 4 days. When 6 tractors work together, each of them ploughs 120 hectares a day. If two of the tractors were moved to another field, then the remaining 4 tractors could plough the same field in 5 days. How many hectares a day would one tractor plough then? Click to see solution Solution: If each of $6$ tractors ploughed $120$ hectares a day and they finished the work in $4$ days, then the whole field is: $120\cdot 6 \cdot 4 = 720 \cdot 4 = 2880$ hectares. Let's suppose that each of the four tractors ploughed $x$ hectares a day. Therefore in 5 days they ploughed $5 \cdot 4 \cdot x = 20 \cdot x$ hectares, which equals the area of the whole field, 2880 hectares. So, we get $20x = 2880$ $ x = \frac{2880}{20} = 144$. Hence, each of the four tractors would plough 144 hectares a day.

Problem 5 A student chose a number, multiplied it by 2, then subtracted 138 from the result and got 102. What was the number he chose? Click to see solution Solution: Let $x$ be the number he chose, then $2\cdot x - 138 = 102$ $2x = 240$ $x = 120$

Problem 6 I chose a number and divide it by 5. Then I subtracted 154 from the result and got 6. What was the number I chose? Click to see solution Solution: Let $x$ be the number I chose, then $\frac{x}{5}-154=6$ $\frac{x}{5}=160$ $x=800$

Problem 8 One side of a rectangle is 3 cm shorter than the other side. If we increase the length of each side by 1 cm, then the area of the rectangle will increase by 18 cm 2 . Find the lengths of all sides. Click to see solution Solution: Let $x$ be the length of the longer side $x \gt 3$, then the other side's length is $x-3$ cm. Then the area is S 1 = x(x - 3) cm 2 . After we increase the lengths of the sides they will become $(x +1)$ and $(x - 3 + 1) = (x - 2)$ cm long. Hence the area of the new rectangle will be $A_2 = (x + 1)\cdot(x - 2)$ cm 2 , which is 18 cm 2 more than the first area. Therefore $A_1 +18 = A_2$ $x(x - 3) + 18 = (x + 1)(x - 2)$ $x^2 - 3x + 18 = x^2 + x - 2x - 2$ $2x = 20$ $x = 10$. So, the sides of the rectangle are $10$ cm and $(10 - 3) = 7$ cm long.

Problem 9 The first year, two cows produced 8100 litres of milk. The second year their production increased by 15% and 10% respectively, and the total amount of milk increased to 9100 litres a year. How many litres were milked from each cow each year? Click to see solution Solution: Let x be the amount of milk the first cow produced during the first year. Then the second cow produced $(8100 - x)$ litres of milk that year. The second year, each cow produced the same amount of milk as they did the first year plus the increase of $15\%$ or $10\%$. So $8100 + \frac{15}{100}\cdot x + \frac{10}{100} \cdot (8100 - x) = 9100$ Therefore $8100 + \frac{3}{20}x + \frac{1}{10}(8100 - x) = 9100$ $\frac{1}{20}x = 190$ $x = 3800$ Therefore, the cows produced 3800 and 4300 litres of milk the first year, and $4370$ and $4730$ litres of milk the second year, respectively.

Problem 10 The distance between stations A and B is 148 km. An express train left station A towards station B with the speed of 80 km/hr. At the same time, a freight train left station B towards station A with the speed of 36 km/hr. They met at station C at 12 pm, and by that time the express train stopped at at intermediate station for 10 min and the freight train stopped for 5 min. Find: a) The distance between stations C and B. b) The time when the freight train left station B. Click to see solution Solution a) Let x be the distance between stations B and C. Then the distance from station C to station A is $(148 - x)$ km. By the time of the meeting at station C, the express train travelled for $\frac{148-x}{80}+\frac{10}{60}$ hours and the freight train travelled for $\frac{x}{36}+\frac{5}{60}$ hours. The trains left at the same time, so: $\frac{148 - x}{80} + \frac{1}{6} = \frac{x}{36} + \frac{1}{12}$. The common denominator for 6, 12, 36, 80 is 720. Then $9(148 - x) +120 = 20x +60$ $1332 - 9x + 120 = 20x + 60$ $29x = 1392$ $x = 48$. Therefore the distance between stations B and C is 48 km. b) By the time of the meeting at station C the freight train rode for $\frac{48}{36} + \frac{5}{60}$ hours, i.e. $1$ hour and $25$ min. Therefore it left station B at $12 - (1 + \frac{25}{60}) = 10 + \frac{35}{60}$ hours, i.e. at 10:35 am.

Problem 11 Susan drives from city A to city B. After two hours of driving she noticed that she covered 80 km and calculated that, if she continued driving at the same speed, she would end up been 15 minutes late. So she increased her speed by 10 km/hr and she arrived at city B 36 minutes earlier than she planned. Find the distance between cities A and B. Click to see solution Solution: Let $x$ be the distance between A and B. Since Susan covered 80 km in 2 hours, her speed was $V = \frac{80}{2} = 40$ km/hr. If she continued at the same speed she would be $15$ minutes late, i.e. the planned time on the road is $\frac{x}{40} - \frac{15}{60}$ hr. The rest of the distance is $(x - 80)$ km. $V = 40 + 10 = 50$ km/hr. So, she covered the distance between A and B in $2 +\frac{x - 80}{50}$ hr, and it was 36 min less than planned. Therefore, the planned time was $2 + \frac{x -80}{50} + \frac{36}{60}$. When we equalize the expressions for the scheduled time, we get the equation: $\frac{x}{40} - \frac{15}{60} = 2 + \frac{x -80}{50} + \frac{36}{60}$ $\frac{x - 10}{40} = \frac{100 + x - 80 + 30}{50}$ $\frac{x - 10}{4} = \frac{x +50}{5}$ $5x - 50 = 4x + 200$ $x = 250$ So, the distance between cities A and B is 250 km.

Problem 12 To deliver an order on time, a company has to make 25 parts a day. After making 25 parts per day for 3 days, the company started to produce 5 more parts per day, and by the last day of work 100 more parts than planned were produced. Find how many parts the company made and how many days this took. Click to see solution Solution: Let $x$ be the number of days the company worked. Then 25x is the number of parts they planned to make. At the new production rate they made: $3\cdot 25 + (x - 3)\cdot 30 = 75 + 30(x - 3)$ Therefore: $25 x = 75 + 30(x -3) - 100$ $25x = 75 +30x -90 - 100$ $190 -75 = 30x -25$ $115 = 5x$ $x = 23$ So the company worked 23 days and they made $23\cdot 25+100 = 675$ pieces.

Problem 13 There are 24 students in a seventh grade class. They decided to plant birches and roses at the school's backyard. While each girl planted 3 roses, every three boys planted 1 birch. By the end of the day they planted $24$ plants. How many birches and roses were planted? Click to see solution Solution: Let $x$ be the number of roses. Then the number of birches is $24 - x$, and the number of boys is $3\times (24-x)$. If each girl planted 3 roses, there are $\frac{x}{3}$ girls in the class. We know that there are 24 students in the class. Therefore $\frac{x}{3} + 3(24 - x) = 24$ $x + 9(24 - x) = 3\cdot 24$ $x +216 - 9x = 72$ $216 - 72 = 8x$ $\frac{144}{8} = x$ $x = 18$ So, students planted 18 roses and 24 - x = 24 - 18 = 6 birches.

Problem 14 A car left town A towards town B driving at a speed of V = 32 km/hr. After 3 hours on the road the driver stopped for 15 min in town C. Because of a closed road he had to change his route, making the trip 28 km longer. He increased his speed to V = 40 km/hr but still he was 30 min late. Find: a) The distance the car has covered. b) The time that took it to get from C to B. Click to see solution Solution: From the statement of the problem we don't know if the 15 min stop in town C was planned or it was unexpected. So we have to consider both cases. A The stop was planned. Let us consider only the trip from C to B, and let $x$ be the number of hours the driver spent on this trip. Then the distance from C to B is $S = 40\cdot x$ km. If the driver could use the initial route, it would take him $x - \frac{30}{60} = x - \frac{1}{2}$ hours to drive from C to B. The distance from C to B according to the initially itinerary was $(x - \frac{1}{2})\cdot 32$ km, and this distance is $28$ km shorter than $40\cdot x$ km. Then we have the equation $(x - 1/2)\cdot 32 + 28 = 40x$ $32x -16 +28 = 40x$ $-8x = -12$ $8x = 12$ $x = \frac{12}{8}$ $x = 1 \frac{4}{8} = 1 \frac{1}{2} = 1 \frac{30}{60} =$ 1 hr 30 min. So, the car covered the distance between C and B in 1 hour and 30 min. The distance from A to B is $3\cdot 32 + \frac{12}{8}\cdot 40 = 96 + 60 = 156$ km. B Suppose it took $x$ hours for him to get from C to B. Then the distance is $S = 40\cdot x$ km. The driver did not plan the stop at C. Let we accept that he stopped because he had to change the route. It took $x - \frac{30}{60} + \frac{15}{60} = x - \frac{15}{60} = x - \frac{1}{4}$ h to drive from C to B. The distance from C to B is $32(x - \frac{1}{4})$ km, which is $28$ km shorter than $40\cdot x$, i.e. $32(x - \frac{1}{4}) + 28 = 40x$ $32x - 8 +28 = 40x$ $20= 8x$ $x = \frac{20}{8} = \frac{5}{2} = 2 \text{hr } 30 \text{min}.$ The distance covered equals $ 40 \times 2.5 = 100 km$.

Problem 15 If a farmer wants to plough a farm field on time, he must plough 120 hectares a day. For technical reasons he ploughed only 85 hectares a day, hence he had to plough 2 more days than he planned and he still has 40 hectares left. What is the area of the farm field and how many days the farmer planned to work initially? Click to see solution Solution: Let $x$ be the number of days in the initial plan. Therefore, the whole field is $120\cdot x$ hectares. The farmer had to work for $x + 2$ days, and he ploughed $85(x + 2)$ hectares, leaving $40$ hectares unploughed. Then we have the equation: $120x = 85(x + 2) + 40$ $35x = 210$ $x = 6$ So the farmer planned to have the work done in 6 days, and the area of the farm field is $120\cdot 6 = 720$ hectares.

Problem 16 A woodworker normally makes a certain number of parts in 24 days. But he was able to increase his productivity by 5 parts per day, and so he not only finished the job in only 22 days but also he made 80 extra parts. How many parts does the woodworker normally makes per day and how many pieces does he make in 24 days? Click to see solution Solution: Let $x$ be the number of parts the woodworker normally makes daily. In 24 days he makes $24\cdot x$ pieces. His new daily production rate is $x + 5$ pieces and in $22$ days he made $22 \cdot (x + 5)$ parts. This is 80 more than $24\cdot x$. Therefore the equation is: $24\cdot x + 80 = 22(x +5)$ $30 = 2x$ $x = 15$ Normally he makes 15 parts a day and in 24 days he makes $15 \cdot 24 = 360$ parts.

Problem 17 A biker covered half the distance between two towns in 2 hr 30 min. After that he increased his speed by 2 km/hr. He covered the second half of the distance in 2 hr 20 min. Find the distance between the two towns and the initial speed of the biker. Click to see solution Solution: Let x km/hr be the initial speed of the biker, then his speed during the second part of the trip is x + 2 km/hr. Half the distance between two cities equals $2\frac{30}{60} \cdot x$ km and $2\frac{20}{60} \cdot (x + 2)$ km. From the equation: $2\frac{30}{60} \cdot x = 2\frac{20}{60} \cdot (x+2)$ we get $x = 28$ km/hr. The intial speed of the biker is 28 km/h. Half the distance between the two towns is $2 h 30 min \times 28 = 2.5 \times 28 = 70$. So the distance is $2 \times 70 = 140$ km.

Problem 18 A train covered half of the distance between stations A and B at the speed of 48 km/hr, but then it had to stop for 15 min. To make up for the delay, it increased its speed by $\frac{5}{3}$ m/sec and it arrived to station B on time. Find the distance between the two stations and the speed of the train after the stop. Click to see solution Solution: First let us determine the speed of the train after the stop. The speed was increased by $\frac{5}{3}$ m/sec $= \frac{5\cdot 60\cdot 60}{\frac{3}{1000}}$ km/hr = $6$ km/hr. Therefore, the new speed is $48 + 6 = 54$ km/hr. If it took $x$ hours to cover the first half of the distance, then it took $x - \frac{15}{60} = x - 0.25$ hr to cover the second part. So the equation is: $48 \cdot x = 54 \cdot (x - 0.25)$ $48 \cdot x = 54 \cdot x - 54\cdot 0.25$ $48 \cdot x - 54 \cdot x = - 13.5$ $-6x = - 13.5$ $x = 2.25$ h. The whole distance is $2 \times 48 \times 2.25 = 216$ km.

Problem 19 Elizabeth can get a certain job done in 15 days, and Tony can finish only 75% of that job within the same time. Tony worked alone for several days and then Elizabeth joined him, so they finished the rest of the job in 6 days, working together. For how many days have each of them worked and what percentage of the job have each of them completed? Click to see solution Solution: First we will find the daily productivity of every worker. If we consider the whole job as unit (1), Elizabeth does $\frac{1}{15}$ of the job per day and Tony does $75\%$ of $\frac{1}{15}$, i.e. $\frac{75}{100}\cdot \frac{1}{15} = \frac{1}{20}$. Suppose that Tony worked alone for $x$ days. Then he finished $\frac{x}{20}$ of the total job alone. Working together for 6 days, the two workers finished $6\cdot (\frac{1}{15}+\frac{1}{20}) = 6\cdot \frac{7}{60} = \frac{7}{10}$ of the job. The sum of $\frac{x}{20}$ and $\frac{7}{10}$ gives us the whole job, i.e. $1$. So we get the equation: $\frac{x}{20}+\frac{7}{10}=1$ $\frac{x}{20} = \frac{3}{10}$ $x = 6$. Tony worked for 6 + 6 = 12 days and Elizabeth worked for $6$ days. The part of job done is $12\cdot \frac{1}{20} = \frac{60}{100} = 60\%$ for Tony, and $6\cdot \frac{1}{15} = \frac{40}{100} = 40\%$ for Elizabeth.

Problem 20 A farmer planned to plough a field by doing 120 hectares a day. After two days of work he increased his daily productivity by 25% and he finished the job two days ahead of schedule. a) What is the area of the field? b) In how many days did the farmer get the job done? c) In how many days did the farmer plan to get the job done? Click to see solution Solution: First of all we will find the new daily productivity of the farmer in hectares per day: 25% of 120 hectares is $\frac{25}{100} \cdot 120 = 30$ hectares, therefore $120 + 30 = 150$ hectares is the new daily productivity. Lets x be the planned number of days allotted for the job. Then the farm is $120\cdot x$ hectares. On the other hand, we get the same area if we add $120 \cdot 2$ hectares to $150(x -4)$ hectares. Then we get the equation $120x = 120\cdot 2 + 150(x -4)$ $x = 12$ So, the job was initially supposed to take 12 days, but actually the field was ploughed in 12 - 2 =10 days. The field's area is $120 \cdot 12 = 1440$ hectares.

Problem 21 To mow a grass field a team of mowers planned to cover 15 hectares a day. After 4 working days they increased the daily productivity by $33 \times \frac{1}{3}\%$, and finished the work 1 day earlier than it was planned. A) What is the area of the grass field? B) How many days did it take to mow the whole field? C) How many days were scheduled initially for this job? Hint : See problem 20 and solve by yourself. Answer: A) 120 hectares; B) 7 days; C) 8 days.

Problem 22 A train travels from station A to station B. If the train leaves station A and makes 75 km/hr, it arrives at station B 48 minutes ahead of scheduled. If it made 50 km/hr, then by the scheduled time of arrival it would still have 40 km more to go to station B. Find: A) The distance between the two stations; B) The time it takes the train to travel from A to B according to the schedule; C) The speed of the train when it's on schedule. Click to see solution Solution: Let $x$ be the scheduled time for the trip from A to B. Then the distance between A and B can be found in two ways. On one hand, this distance equals $75(x - \frac{48}{60})$ km. On the other hand, it is $50x + 40$ km. So we get the equation: $75(x - \frac{48}{60}) = 50x + 40$ $x = 4$ hr is the scheduled travel time. The distance between the two stations is $50\cdot 4 +40 = 240$ km. Then the speed the train must keep to be on schedule is $\frac{240}{4} = 60$ km/hr.

Problem 23 The distance between towns A and B is 300 km. One train departs from town A and another train departs from town B, both leaving at the same moment of time and heading towards each other. We know that one of them is 10 km/hr faster than the other. Find the speeds of both trains if 2 hours after their departure the distance between them is 40 km. Click to see solution Solution: Let the speed of the slower train be $x$ km/hr. Then the speed of the faster train is $(x + 10)$ km/hr. In 2 hours they cover $2x$ km and $2(x +10)$km, respectively. Therefore if they didn't meet yet, the whole distance from A to B is $2x + 2(x +10) +40 = 4x +60$ km. However, if they already met and continued to move, the distance would be $2x + 2(x + 10) - 40 = 4x - 20$km. So we get the following equations: $4x + 60 = 300$ $4x = 240$ $x = 60$ or $4x - 20 = 300$ $4x = 320$ $x = 80$ Hence the speed of the slower train is $60$ km/hr or $80$ km/hr and the speed of the faster train is $70$ km/hr or $90$ km/hr.

Problem 24 A bus travels from town A to town B. If the bus's speed is 50 km/hr, it will arrive in town B 42 min later than scheduled. If the bus increases its speed by $\frac{50}{9}$ m/sec, it will arrive in town B 30 min earlier than scheduled. Find: A) The distance between the two towns; B) The bus's scheduled time of arrival in B; C) The speed of the bus when it's on schedule. Click to see solution Solution: First we will determine the speed of the bus following its increase. The speed is increased by $\frac{50}{9}$ m/sec $= \frac{50\cdot60\cdot60}{\frac{9}{1000}}$ km/hr $= 20$ km/hr. Therefore, the new speed is $V = 50 + 20 = 70$ km/hr. If $x$ is the number of hours according to the schedule, then at the speed of 50 km/hr the bus travels from A to B within $(x +\frac{42}{60})$ hr. When the speed of the bus is $V = 70$ km/hr, the travel time is $x - \frac{30}{60}$ hr. Then $50(x +\frac{42}{60}) = 70(x-\frac{30}{60})$ $5(x+\frac{7}{10}) = 7(x-\frac{1}{2})$ $\frac{7}{2} + \frac{7}{2} = 7x -5x$ $2x = 7$ $x = \frac{7}{2}$ hr. So, the bus is scheduled to make the trip in $3$ hr $30$ min. The distance between the two towns is $70(\frac{7}{2} - \frac{1}{2}) = 70\cdot 3 = 210$ km and the scheduled speed is $\frac{210}{\frac{7}{2}} = 60$ km/hr.

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High school geometry

Unit 1: performing transformations, unit 2: transformation properties and proofs, unit 3: congruence, unit 4: similarity, unit 5: right triangles & trigonometry, unit 6: analytic geometry, unit 7: conic sections, unit 8: circles, unit 9: solid geometry, review articles.