Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 3 • Question 16

In a shop, 4 T-shirts and 2 jumpers cost £81.80, and 3 T-shirts and 3 jumpers cost £92.10. Work out the individual cost of a T-shirt and a jumper.

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Video Transcript

In a shop, four T-shirts and two jumpers cost 81 pounds 80 and three T-shirts and three jumpers cost 92 pounds 10. Work out the individual cost of a T-shirt and a jumper.

To answer this question, we’re going to need to use algebra. And we’re going to need to translate the information we’ve been given into a pair of equations. First, we’ll introduce some letters. And it doesn’t really matter which letters we use. But perhaps, in this question, it makes sense to use 𝑡 to represent the cost of one T-shirt and 𝑗 to represent the cost of one jumper.

Let’s look at the first statement we’re given. Four T-shirts, so that’s four 𝑡, and two jumpers, so that’s plus two 𝑗, cost 81 pounds 80 which we can write as 81.8. So this gives us our first equation. Now, let’s look at the second statement we’re given. Three T-shirts — so that’s three 𝑡 — and three jumpers — plus three 𝑗 — cost 92 pounds 10 which you can write as 92.1. So now, we have our second equation.

What we’ve now created are a pair of linear simultaneous equations in 𝑡 and 𝑗, which we need to solve to find each of their values. We’ll answer this question using the elimination method. And what we want to do first of all is manipulate each of these equations. That’s just multiplying or divide them by some number so that we end up either the same number of 𝑡 or the same number of 𝑗 in the two equations.

There are various different ways that we could do this. But let’s consider equation one first. And we can actually divide every term in equation one by two. Four 𝑡 divided by two gives two 𝑡. Two 𝑗 divided by two just gives one 𝑗 or 𝑗. And 81.8 divided by two gives 40.9. This equation is equivalent to equation one because we’ve done the same thing to every term. But it’s slightly simpler because the coefficient of 𝑗 is just one. We can label this equation as equation three.

Now let’s consider equation two. We can divide every term in this equation by three. Three 𝑡 divided by three is 𝑡. Three 𝑗 divided by three is 𝑗. And 92.1 divided by three is 30.7. Again, this equation is exactly equivalent to equation two as we did the same thing to all three terms. But it’s slightly simpler. And again, we have a coefficient of 𝑗 which is just one which now matches the coefficient of 𝑗 in equation three.

We can now subtract equation four from equation three. Two 𝑡 minus 𝑡 is one 𝑡 which we just write as 𝑡. And 𝑗 minus 𝑗 is zero. The terms cancel out. So we’re left with an equation in 𝑡 only. On the right, we have 40.9 minus 30.7 which is 10.2. We found then the value of 𝑡. It’s 10.2.

Remember though that this represented the cost of a T-shirt. So it needs to be in pounds. And we need two decimal places. So we can add a zero in the second decimal place. We found that the cost of a T-shirt is 10 pounds 20.

To find the value of 𝑗, we can then substitute the value of 𝑡 into any of our four equations. I’m going to choose to substitute into equation four. Equation four was 𝑡 plus 𝑗 equals 30.7. So now we have 10.2 plus 𝑗 equals 30.7. To solve this equation for 𝑗, we need to subtract 10.2 from each side. And it gives 𝑗 equals 20.5.

Again, we need to write this as a value with the pound sign and with a zero in the second decimal place. So we found that the cost of a jumper is 20 pounds 50.

It’s always good to check our answers where we can. So we can substitute the values we’ve found for 𝑡 and 𝑗 back into any of these equations. I’m going to substitute into equation one. The left-hand side of equation one was four 𝑡 plus two 𝑗. So substituting 10.2 for 𝑡 and 20.5 for 𝑗, we have four multiplied by 10.2 plus two multiplied by 20.5. This gives 40.8 plus 41 which is equal to 81.8. This is equal to the value on the right-hand side of equation one and the same as the cost of four T-shirts and two jumpers.

So this confirms that our solution is correct. You could also calculate the cost of three T-shirts and three jumpers if you wish to again check the answer. We found that the individual cost of one T-shirt is 10 pounds 20 and the cost of one jumper is 20 pounds 50.

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