Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 15

𝑥 × 10⁵ + 𝑥 × 10² = 123123, where 𝑥 is a number. Calculate 𝑥 × 10⁵ − 𝑥 × 10², giving your answer in standard form.

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Video Transcript

𝑥 multiplied by 10 to the power of five plus 𝑥 multiplied by 10 squared is equal to 123123, where 𝑥 is a number. Calculate 𝑥 multiplied by 10 to the power of five minus 𝑥 multiplied by 10 to the power of two, giving your answer in standard form.

This does look really tricky. But the important thing to spot is that 𝑥 is the same in both parts of this sum. Since 𝑥 is the same, we know it must have the digits one, two, and three. But how do we decide whether 𝑥 is 123, 12.3, 1.23, or even 0.123? Well, these numbers are in standard form. When numbers are in standard form, we write them as some number multiplied by 10 to the power of some other number.

The power represented by the letter 𝑏 in this example is an integer, is a whole number. And 𝑎 has to be greater than or equal to one and less than 10. We should assume then that 𝑥 must be equal to 1.23 since it satisfies these conditions. Let’s check what that means for the first number: 𝑥 multiplied by 10 to the power of five is 1.23 multiplied by 10 to the power of five.

10 to the power of five is 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 which is 100000. So 1.23 multiplied by 10 to the power of five is 1.23 multiplied by 100000 which is 123000. That also means that 𝑥 multiplied by 10 squared is 1.23 multiplied by 10 squared. 10 squared is 10 multiplied by itself. So it’s 100. And that means that 1.23 multiplied by 100 is 123. If we add these two values 123000 and 123, we actually get the answer we need: 123123.

To answer this question then, we are going to need to subtract the two values. We’re going to subtract 123 from 123000. Let’s consider the column method. We can’t do zero take away three. So we borrow from the first digit to the left which is not zero. Here that’s three. When we borrow from the three, it becomes two and the digit immediately to its right becomes 10.

We then borrow from the 10 and it becomes nine and the digit immediately to its right becomes 10. Then again, we borrow from that 10 and it becomes nine. And the digit that we’re interested in — the one on the furthest to the right — is 10. 10 minus three is seven. Nine minus two is seven. Nine minus one is eight. And then, two minus zero is two. Two minus zero is two. And one minus zero is one.

And this means that 𝑥 multiplied by 10 to the power of five minus 𝑥 multiplied by 10 squared is 122877. But we’re not quite finished. We need to give our answer in standard form. Remember we said that when we put a number in standard form, whatever we multiply by 10 to some power must be between one and 10. So our number is going to become 1.22877 multiplied by 10 to some integer.

To work out what this integer is, we need to figure out how many times we need to multiply 1.22877 by 10 to get our original number. In fact, we need to move the digits to the left five spaces. That’s the same as multiplying by 10 to the power of five. That means that our answer is 1.22877 multiplied by 10 to the power of five.

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