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Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

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How can i solve error 0xc004f074?

i cant activate my product key. when i enter settings, i cant change many things then at the top it appers windows key is not active. it tooks me into System>Activation here i cant activete it and the Error 0xc004f074 appers. i learned and did everthing about this problem but still could not solve it.

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• Microsoft Agent |

Hello and welcome to Microsoft Community!

I understand that you are experiencing 0xc004f074 when activating your product key, and I think I need to know the source of your product key, which will allow me to better help you troubleshoot the problem.

In general, this error code may indicate:

0xC004F074 and its description "Key Management Server (KMS) not available"

Error: 0xC004F074 Software License Service reported unable to activate the product. No Key Management Service (KMS) could be contacted.

It is also possible that the system files are corrupt.

Next I will provide my method to help you try to fix the error.

Using the slmgr.vbs command

1. Press Windows Key + S, type cmd , and then select "Run as administrator" at the command prompt.

2. Type the following command (replace the X in the code with your 25-digit product key number) and click Enter

slmgr.vbs -ipk XXXXX - XXXXX -XXXXX -XXXXX -XXXXX

3. Type the following command and click Enter

slmg.vbs -ato

4. Restart your computer and see if your device shows the same error. 

To run an SFC scan

1. Open a command prompt with administrative privileges.

2. Type the following command and hit Enter

sfc/scannow

3. Wait for the command to finish running and then restart your computer, which may replace and fix the error code 0xc004f074 System files without a product key but with a digital license that may cause failure to activate Windows.

Run Update and Activation Troubleshooting

1. Press Windows Key + I to open the Settings application, and then select Update and Security.

2. Click Troubleshooting in the left pane and select Other Troubleshooting.

3. Now, select Windows Update and click the Run Troubleshooting button.

4. If this does not solve the problem, go back to the Update and Security window and select Activate in the left pane.

5. Finally, click on the Troubleshooting option and apply any recommendations from the tool.

On the other hand, Windows Activation Troubleshooting is the best option for most activation issues.

————————————————————————————————————

These are some of the suggestions I can offer, I hope my suggestions can help you.

I'm looking forward to your feedback!

Best Regards!

Leo - Microsoft Community Support Specialist

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How Biden Adopted Trump’s Trade War With China

The president has proposed new barriers to electric vehicles, steel and other goods..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Sabrina Tavernise, and this is “The Daily.”

[MUSIC PLAYING]

Donald Trump upended decades of American policy when he started a trade war with China. Many thought that President Biden would reverse those policies. Instead, he’s stepping them up. Today, my colleague, Jim Tankersley, explains.

It’s Monday, May 13.

Jim, it’s very nice to have you in the studio.

It’s so great to be here, Sabrina. Thank you so much.

So we are going to talk today about something I find very interesting and I know you’ve been following. We’re in the middle of a presidential campaign. You are an economics reporter looking at these two candidates, and you’ve been trying to understand how Trump and Biden are thinking about our number one economic rival, and that is China.

As we know, Trump has been very loud and very clear about his views on China. What about Biden?

Well, no one is going to accuse President Biden of being as loud as former President Trump. But I think he’s actually been fairly clear in a way that might surprise a lot of people about how he sees economic competition with China.

We’re going after China in the wrong way. China is stealing intellectual property. China is conditioning —

And Biden has, kind of surprisingly, sounded a lot, in his own Joe Biden way, like Trump.

They’re not competing. They’re cheating. They’re cheating. And we’ve seen the damage here in America.

He has been very clear that he thinks China is cheating in trade.

The bottom line is I want fair competition with China, not conflict. And we’re in a stronger position to win the economic competition of the 21st century against China or anyone else because we’re investing in America and American workers again. Finally.

And maybe the most surprising thing from a policy perspective is just how much Biden has built on top of the anti-China moves that Trump made and really is the verge of his own sort of trade war with China.

Interesting. So remind us, Jim, what did Trump do when he actually came into office? We, of course, remember Trump really talking about China and banging that drum hard during the campaign, but remind us what he actually did when he came into office.

Yeah, it’s really instructive to start with the campaign, because Trump is talking about China in some very specific ways.

We have a $500 billion deficit, trade deficit, with China. We’re going to turn it around. And we have the cards. Don’t forget —

They’re ripping us off. They’re stealing our jobs.

They’re using our country as a piggy bank to rebuild China, and many other countries are doing the same thing. So we’re losing our good jobs, so many.

The economic context here is the United States has lost a couple of million jobs in what was called the China shock of the early 2000s. And Trump is tapping into that.

But when the Chinese come in, and they want to make great trade deals — and they make the best trade deals, and not anymore. When I’m there, we turn it around, folks. We turn it around. We have —

And what he’s promising as president is that he’s going to bring those jobs back.

I’ll be the greatest jobs president that God ever created. I’ll take them back from China, from Japan.

And not just any jobs, good-paying manufacturing jobs, all of it — clothes, shoes, steel, all of these jobs that have been lost that American workers, particularly in the industrial Midwest, used to do. Trump’s going to bring them back with policy meant to rebalance the trade relationship with China to get a better deal with China.

So he’s saying China is eating our lunch and has been for decades. That’s the reason why factory workers in rural North Carolina don’t have work. It’s those guys. And I’m going to change that.

Right. And he likes to say it’s because our leaders didn’t cut the right deal with them, so I’m going to make a better deal. And to get a better deal, you need leverage. So a year into his presidency, he starts taking steps to amass leverage with China.

And so what does that look like?

Just an hour ago, surrounded by a hand-picked group of steelworkers, President Trump revealed he was not bluffing.

It starts with tariffs. Tariffs are taxes that the government imposes on imports.

Two key global imports into America now face a major new barrier.

Today, I’m defending America’s national security by placing tariffs on foreign imports of steel and aluminum.

And in this case, it’s imports from a lot of different countries, but particularly China.

Let’s take it straight to the White House. The president of the United States announcing new trade tariffs against China. Let’s listen in.

This has been long in the making. You’ve heard —

So Trump starts, in 2018, this series of tariffs that he’s imposing on all sorts of things — washing machines, solar panels, steel, aluminum. I went to Delaware to a lighting store at that time, I remember, where basically everything they sold came from China and was subject to the Trump tariffs, because that’s where lighting was made now.

Interesting.

Hundreds of billions of dollars of Chinese goods now start falling under these Trump tariffs. The Chinese, of course, don’t take this lying down.

China says it is not afraid of a trade war with the US, and it’s fighting back against President Trump with its own tariffs on US goods.

They do their own retaliatory tariffs. Now American exports to China cost more for Chinese consumers. And boom, all of a sudden, we are in the midst of a full-blown trade war between the United States and Beijing.

Right. And that trade war was kind of a shock because for decades, politicians had avoided that kind of policy. It was the consensus of the political class in the United States that there should not be tariffs like that. It should be free trade. And Trump just came in and blew up the consensus.

Yeah. And Sabrina, I may have mentioned this once or 700 times before on this program, but I talk to a lot of economists in my job.

Yeah, it’s weird. I talk to a lot of economists. And in 2018 when this started, there were very, very, very few economists of any political persuasion who thought that imposing all these tariffs were a good idea. Republican economists in particular, this is antithetical to how they think about the world, which is low taxes, free trade. And even Democratic economists who thought they had some problems with the way free trade had been conducted did not think that Trump’s “I’m going to get a better deal” approach was going to work. And so there was a lot of criticism at the time, and a lot of politicians really didn’t like it, a lot of Democrats, many Republicans. And it all added up to just a real, whoa, I don’t think this is going to work.

So that begs the question, did it?

Well, it depends on what you mean by work. Economically, it does not appear to have achieved what Trump wanted. There’s no evidence yet in the best economic research that’s been done on this that enormous amounts of manufacturing jobs came back to the United States because of Trump’s tariffs. There was research, for example, on the tariffs on washing machines. They appear to have helped a couple thousand jobs, manufacturing jobs be created in the United States, but they also raised the price of washing machines for everybody who bought them by enough that each additional job that was created by those tariffs effectively cost consumers, like, $800,000 per job.

There’s like lots of evidence that the sectors Trump was targeting to try to help here, he didn’t. There just wasn’t a lot of employment rebound to the United States. But politically, it really worked. The tariffs were very popular. They had this effect of showing voters in those hollowed-out manufacturing areas that Trump was on their team and that he was fighting for them. Even if they didn’t see the jobs coming back, they felt like he was standing up for them.

So the research suggests this was a savvy political move by Trump. And in the process, it sort of changes the political economic landscape in both parties in the United States.

Right. So Trump made these policies that seemed, for many, many years in the American political system, fringe, isolationist, economically bad, suddenly quite palatable and even desirable to mainstream policymakers.

Yeah. Suddenly getting tough on China is something everyone wants to do across both parties. And so from a political messaging standpoint, being tough on China is now where the mainstream is. But at the same time, there is still big disagreement over whether Trump is getting tough on China in the right way, whether he’s actually being effective at changing the trade relationship with China.

Remember that Trump was imposing these tariffs as a way to get leverage for a better deal with China. Well, he gets a deal of sorts, actually, with the Chinese government, which includes some things about tariffs, and also China agreeing to buy some products from the United States. Trump spins it as this huge win, but nobody else really, including Republicans, acts like Trump has solved the problem that Trump himself has identified. This deal is not enough to make everybody go, well, everything’s great with China now. We can move on to the next thing.

China remains this huge issue. And the question of what is the most effective way to deal with them is still an animating force in politics.

Got it. So politically, huge win, but policy-wise and economically, and fundamentally, the problem of China still very much unresolved.

Absolutely.

So then Biden comes in. What does Biden do? Does he keep the tariffs on?

Biden comes to office, and there remains this real pressure from economists to roll back what they consider to be the ineffective parts of Trump’s trade policy. That includes many of the tariffs. And it’s especially true at a time when almost immediately after Biden takes office, inflation spikes. And so Americans are paying a lot of money for products, and there’s this pressure on Biden, including from inside his administration, to roll back some of the China tariffs to give Americans some relief on prices.

And Biden considers this, but he doesn’t do it. He doesn’t reverse Trump’s tariff policy. In the end, he’s actually building on it.

We’ll be right back.

So Jim, you said that Biden is actually building on Trump’s anti-China policy. What exactly does that look like?

So Biden builds on the Trump China policy in three key ways, but he does it with a really specific goal that I just want you to keep in mind as we talk about all of this, which is that Biden isn’t just trying to beat China on everything. He’s not trying to cut a better deal. Biden is trying to beat China in a specific race to own the clean-energy future.

Clean energy.

Yeah. So keep that in mind, clean energy. And the animating force behind all of the things Biden does with China is that Biden wants to beat China on what he thinks are the jobs of the future, and that’s green technology.

Got it. OK. So what does he do first?

OK. Thing number one — let’s talk about the tariffs. He does not roll them back. And actually, he builds on them. For years, for the most part, he just lets the tariffs be. His administration reviews them. And it’s only now, this week, when his administration is going to actually act on the tariffs. And what they’re going to do is raise some of them. They’re going to raise them on strategic green tech things, like electric vehicles, in order to make them more expensive.

And I think it’s important to know the backdrop here, which is since Biden has taken office, China has started flooding global markets with really low-cost green technologies. Solar panels, electric vehicles are the two really big ones. And Biden’s aides are terrified that those imports are going to wash over the United States and basically wipe out American automakers, solar panel manufacturers, that essentially, if Americans can just buy super-cheap stuff from China, they’re not going to buy it from American factories. Those factories are going to go out of business.

So Biden’s goal of manufacturing jobs in clean energy, China is really threatening that by dumping all these products on the American market.

Exactly. And so what he wants to do is protect those factories with tariffs. And that means increasing the tariffs that Trump put on electric vehicles in hopes that American consumers will find them too expensive to buy.

But doesn’t that go against Biden’s goal of clean energy and things better for the environment? Lots of mass-market electric vehicles into the United States would seem to advance that goal. And here, he’s saying, no, you can’t come in.

Right, because Biden isn’t just trying to reduce emissions at all costs. He wants to reduce emissions while boosting American manufacturing jobs. He doesn’t want China to get a monopoly in these areas. And he’s also, in particular, worried about the politics of lost American manufacturing jobs. So Biden does not want to just let you buy cheaper Chinese technologies, even if that means reducing emissions.

He wants to boost American manufacturing of those things to compete with China, which brings us to our second thing that Biden has done to build on Trump’s China policy, which is that Biden has started to act like the Chinese government in particular areas by showering American manufacturers with subsidies.

I see. So dumping government money into American businesses.

Yes, tax incentives, direct grants. This is a way that China has, in the past decades, built its manufacturing dominance, is with state support for factories. Biden is trying to do that in particular targeted industries, including electric vehicles, solar power, wind power, semiconductors. Biden has passed a bunch of legislation that showers those sectors with incentives and government support in hopes of growing up much faster American industry.

Got it. So basically, Biden is trying to beat China at its own game.

Yeah, he’s essentially using tariffs to build a fortress around American industry so that he can train the troops to fight the clean energy battle with China.

And the troops being American companies.

Yes. It’s like, we’re going to give them protection — protectionist policy — in order to get up to size, get up to strength as an army in this battle for clean energy dominance against the Chinese.

Got it. So he’s trying to build up the fortress. What’s the third thing Biden does? You mentioned three things.

Biden does not want the United States going it alone against China. He’s trying to build an international coalition, wealthy countries and some other emerging countries that are going to take on China and try to stop the Chinese from using their trade playbook to take over all these new emerging industrial markets.

But, Jim, why? What does the US get from bringing our allies into this trade war? Why does the US want that?

Some of this really is about stopping China from gaining access to new markets. It’s like, if you put the low-cost Chinese exports on a boat, and it’s going around the world, looking for a dock to stop and offload the stuff and sell it, Biden wants barriers up at every possible port. And he wants factories in those places that are competing with the Chinese.

And a crucial fact to know here is that the United States and Europe, they are behind China when it comes to clean-energy technology. The Chinese government has invested a lot more than America and Europe in building up its industrial capacity for clean energy. So America and its allies want to deny China dominance of those markets and to build up their own access to them.

And they’re behind, so they’ve got to get going. It’s like they’re in a race, and they’re trailing.

Yeah, it’s an economic race to own these industries, and it’s that global emissions race. They also want to be bringing down fossil-fuel emissions faster than they currently are, and this is their plan.

So I guess, Jim, the question in my mind is, Trump effectively broke the seal, right? He started all of these tariffs. He started this trade war with China. But he did it in this kind of jackhammer, non-targeted way, and it didn’t really work economically. Now Biden is taking it a step further. But the question is, is his effort here going to work?

The answer to whether it’s going to work really depends on what your goals are. And Biden and Trump have very different goals. If Trump wins the White House back, he has made very clear that his goal is to try to rip the United States trade relationship with China even more than he already has. He just wants less trade with China and more stuff of all types made in the United States that used to be made in China. That’s a very difficult goal, but it’s not Biden’s goal.

Biden’s goal is that he wants America to make more stuff in these targeted industries. And there is real skepticism from free-market economists that his industrial policies will work on that, but there’s a lot of enthusiasm for it from a new strain of Democratic economists, in particular, who believe that the only chance Biden has to make that work is by pulling all of these levers, by doing the big subsidies and by putting up the tariffs, that you have to have both the troops training and the wall around them. And if it’s going to work, he has to build on the Trump policies. And so I guess you’re asking, will it work? It may be dependent upon just how far he’s willing to go on the subsidies and the barriers.

There’s a chance of it.

So, Jim, at the highest level, whatever the economic outcome here, it strikes me that these moves by Biden are pretty remarkably different from the policies of the Democratic Party over the decades, really going in the opposite direction. I’m thinking of Bill Clinton and NAFTA in the 1990s. Free trade was the real central mantra of the Democratic Party, really of both parties.

Yeah, and Biden is a real break from Clinton. And Clinton was the one who actually signed the law that really opened up trade with China, and Biden’s a break from that. He’s a break from even President Obama when he was vice president. Biden is doing something different. He’s breaking from that Democratic tradition, and he’s building on what Trump did, but with some throwback elements to it from the Roosevelt administration and the Eisenhower administration. This is this grand American tradition of industrial policy that gave us the space race and the interstate highway system. It’s the idea of using the power of the federal government to build up specific industrial capacities. It was in vogue for a time. It fell out of fashion and was replaced by this idea that the government should get out of the way, and you let the free market drive innovation. And now that industrial policy idea is back in vogue, and Biden is doing it.

So it isn’t just a shift or an evolution. It’s actually a return to big government spending of the ‘30s and the ‘40s and the ‘50s of American industrialism of that era. So what goes around comes around.

Yeah, and it’s a return to that older economic theory with new elements. And it’s in part because of the almost jealousy that American policymakers have of China and the success that it’s had building up its own industrial base. But it also has this political element to it. It’s, in part, animated by the success that Trump had making China an issue with working-class American voters.

You didn’t have to lose your job to China to feel like China was a stand-in for the forces that have taken away good-paying middle-class jobs from American workers who expected those jobs to be there. And so Trump tapped into that. And Biden is trying to tap into that. And the political incentives are pushing every future American president to do more of that. So I think we are going to see even more of this going forward, and that’s why we’re in such an interesting moment right now.

So we’re going to see more fortresses.

More fortresses, more troops, more money.

Jim, thank you.

You’re welcome.

Here’s what else you should know today. Intense fighting between Hamas fighters and Israeli troops raged in parts of Northern Gaza over the weekend, an area where Israel had declared Hamas defeated earlier in the war, only to see the group reconstitute in the power vacuum that was left behind. The persistent lawlessness raised concerns about the future of Gaza among American officials. Secretary of State Antony Blinken said on “Face the Nation” on Sunday that the return of Hamas to the North left him concerned that Israeli victories there would be, quote, “not sustainable,” and said that Israel had not presented the United States with any plan for when the war ends.

And the United Nations aid agency in Gaza said early on Sunday that about 300,000 people had fled from Rafah over the past week, the city in the enclave’s southernmost tip where more than a million displaced Gazans had sought shelter from Israeli bombardments elsewhere. The UN made the announcement hours after the Israeli government issued new evacuation orders in Rafah, deepening fears that the Israeli military was preparing to invade the city despite international warnings.

Today’s episode was produced by Nina Feldman, Carlos Prieto, Sidney Harper, and Luke Vander Ploeg. It was edited by M.J. Davis Lin, Brendan Klinkenberg, and Lisa Chow. Contains original music by Diane Wong, Marion Lozano, and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Produced by Nina Feldman ,  Carlos Prieto ,  Sydney Harper and Luke Vander Ploeg

Edited by M.J. Davis Lin ,  Brendan Klinkenberg and Lisa Chow

Original music by Diane Wong ,  Marion Lozano and Dan Powell

Engineered by Alyssa Moxley

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Donald Trump upended decades of American policy when he started a trade war with China. Many thought that President Biden would reverse those policies. Instead, he’s stepping them up.

Jim Tankersley, who covers economic policy at the White House, explains.

On today’s episode

Jim Tankersley , who covers economic policy at the White House for The New York Times.

Background reading

Mr. Biden, competing with Mr. Trump to be tough on China , called for steel tariffs last month.

The Biden administration may raise tariffs on electric vehicles from China to 100 percent .

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Facility for Rare Isotope Beams

At michigan state university, international research team uses wavefunction matching to solve quantum many-body problems, new approach makes calculations with realistic interactions possible.

FRIB researchers are part of an international research team solving challenging computational problems in quantum physics using a new method called wavefunction matching. The new approach has applications to fields such as nuclear physics, where it is enabling theoretical calculations of atomic nuclei that were previously not possible. The details are published in Nature (“Wavefunction matching for solving quantum many-body problems”) .

Ab initio methods and their computational challenges

An ab initio method describes a complex system by starting from a description of its elementary components and their interactions. For the case of nuclear physics, the elementary components are protons and neutrons. Some key questions that ab initio calculations can help address are the binding energies and properties of atomic nuclei not yet observed and linking nuclear structure to the underlying interactions among protons and neutrons.

Yet, some ab initio methods struggle to produce reliable calculations for systems with complex interactions. One such method is quantum Monte Carlo simulations. In quantum Monte Carlo simulations, quantities are computed using random or stochastic processes. While quantum Monte Carlo simulations can be efficient and powerful, they have a significant weakness: the sign problem. The sign problem develops when positive and negative weight contributions cancel each other out. This cancellation results in inaccurate final predictions. It is often the case that quantum Monte Carlo simulations can be performed for an approximate or simplified interaction, but the corresponding simulations for realistic interactions produce severe sign problems and are therefore not possible.

Using ‘plastic surgery’ to make calculations possible

The new wavefunction-matching approach is designed to solve such computational problems. The research team—from Gaziantep Islam Science and Technology University in Turkey; University of Bonn, Ruhr University Bochum, and Forschungszentrum Jülich in Germany; Institute for Basic Science in South Korea; South China Normal University, Sun Yat-Sen University, and Graduate School of China Academy of Engineering Physics in China; Tbilisi State University in Georgia; CEA Paris-Saclay and Université Paris-Saclay in France; and Mississippi State University and the Facility for Rare Isotope Beams (FRIB) at Michigan State University (MSU)—includes  Dean Lee , professor of physics at FRIB and in MSU’s Department of Physics and Astronomy and head of the Theoretical Nuclear Science department at FRIB, and  Yuan-Zhuo Ma , postdoctoral research associate at FRIB.

“We are often faced with the situation that we can perform calculations using a simple approximate interaction, but realistic high-fidelity interactions cause severe computational problems,” said Lee. “Wavefunction matching solves this problem by doing plastic surgery. It removes the short-distance part of the high-fidelity interaction, and replaces it with the short-distance part of an easily computable interaction.”

This transformation is done in a way that preserves all of the important properties of the original realistic interaction. Since the new wavefunctions look similar to that of the easily computable interaction, researchers can now perform calculations using the easily computable interaction and apply a standard procedure for handling small corrections called perturbation theory.  A team effort

The research team applied this new method to lattice quantum Monte Carlo simulations for light nuclei, medium-mass nuclei, neutron matter, and nuclear matter. Using precise ab initio calculations, the results closely matched real-world data on nuclear properties such as size, structure, and binding energies. Calculations that were once impossible due to the sign problem can now be performed using wavefunction matching.

“It is a fantastic project and an excellent opportunity to work with the brightest nuclear scientist s in FRIB and around the globe,” said Ma. “As a theorist , I'm also very excited about programming and conducting research on the world's most powerful exascale supercomputers, such as Frontier , which allows us to implement wavefunction matching to explore the mysteries of nuclear physics.”

While the research team focused solely on quantum Monte Carlo simulations, wavefunction matching should be useful for many different ab initio approaches, including both classical and  quantum computing calculations. The researchers at FRIB worked with collaborators at institutions in China, France, Germany, South Korea, Turkey, and United States.

“The work is the culmination of effort over many years to handle the computational problems associated with realistic high-fidelity nuclear interactions,” said Lee. “It is very satisfying to see that the computational problems are cleanly resolved with this new approach. We are grateful to all of the collaboration members who contributed to this project, in particular, the lead author, Serdar Elhatisari.”

This material is based upon work supported by the U.S. Department of Energy, the U.S. National Science Foundation, the German Research Foundation, the National Natural Science Foundation of China, the Chinese Academy of Sciences President’s International Fellowship Initiative, Volkswagen Stiftung, the European Research Council, the Scientific and Technological Research Council of Turkey, the National Natural Science Foundation of China, the National Security Academic Fund, the Rare Isotope Science Project of the Institute for Basic Science, the National Research Foundation of Korea, the Institute for Basic Science, and the Espace de Structure et de réactions Nucléaires Théorique.

Michigan State University operates the Facility for Rare Isotope Beams (FRIB) as a user facility for the U.S. Department of Energy Office of Science (DOE-SC), supporting the mission of the DOE-SC Office of Nuclear Physics. Hosting what is designed to be the most powerful heavy-ion accelerator, FRIB enables scientists to make discoveries about the properties of rare isotopes in order to better understand the physics of nuclei, nuclear astrophysics, fundamental interactions, and applications for society, including in medicine, homeland security, and industry.

The U.S. Department of Energy Office of Science is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of today’s most pressing challenges. For more information, visit energy.gov/science.

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