# Video: Pack 2 • Paper 3 • Question 10

Pack 2 • Paper 3 • Question 10

08:15

### Video Transcript

Jim is buying an electric drill. The cost of the drill at the local DIY shop is 139 pounds 50. Jim notices the drill has been discounted by 22.5 percent of its original value. Calculate the original price of the drill.

Now we need to be really careful here because the discount has already been applied. And we’re asked to work backwards to find the original price of the drill. We’ll use a bit of algebra to help us answer this question. And we’ll let 𝑥 represent the original price of the drill.

If the cost of the drill has been discounted by 22.5 percent, then this means that the price we’re given represents 77.5 percent of the original price, as 77.5 is 100 minus 22.5. Let’s think about how the discount cost of the drill would have been calculated.

To find 77.5 percent of an amount, we can divide that amount by 100 to find one percent and then multiply by 77.5. This is equivalent to multiplying by the fraction 77.5 over 100, which is equivalent to multiplying by the decimal 0.775.

So the original cost of the drill 𝑥 was multiplied by 0.775 in order to give the discount price of 139 pounds 50. This gives an equation that we can solve in order to find the original price of the drill. We need to divide both sides of the equation by 0.775, giving 𝑥 is equal to 139.5, or five zero, divided by 0.775. This gives 180. So the original price of the drill is 180 pounds.

Now you do need to be really careful with questions like this. This is an example of a reverse percentage calculation, where the percentage change has already been applied and we’re working backwards to find the original amount. It’s a type of percentage question on which errors are most frequently made.

The most common mistake is to think that, in order to get back to the original amount, we need to add in this case 22.5 percent of the discount cost back on. This would give the calculation 1.225 multiplied by 139.5, as we’ll be finding 122.5 percent of the discounted cost.

This, however, gives 170.8875, which is not the same as the answer we’ve calculated. The reason for this was that the amount subtracted from the original cost was 22.5 percent of the original cost, whereas here we’re adding on 22.5 percent of the discounted cost. And those aren’t the same amount. As I said, this is a really common error. So learn the correct way to answer a reverse percentage calculation and don’t make this mistake.

When Jim bought his house, it had a market value of 300000 pounds. For the first three years that Jim owned his house, the market value of all houses increased by 𝑥 percent each year. In the fourth year that Jim owned his house, the market value of all houses stayed the same. Jim increased the value of his house by 3.2 percent through a number of DIY projects. At the end of the four years, Jim’s house was worth 352291 pounds and 72 pence. Work out 𝑥, the rate at which the market price of houses increased for the first three years that Jim owned his house.

Okay, so there’s a lot of information given here. The first piece is that the original market value of Jim’s house was 300000 pounds. The simpler percentage increase is what happens in the fourth year. The market value of all houses stayed the same. But Jim increased the value of his house by 3.2 percent. This means that, at the end of the fourth year, Jim’s house would be worth 103.2 percent of what it was worth at the beginning of that year or the end of the third year. To find 103.2 percent of an amount, we can divide it by 100 to find one percent and then multiply by 103.2. This will be equivalent to multiplying by the decimal 1.032.

Now let’s look back at the first three years where the market value of all houses, including Jim’s, increased by 𝑥 percent each year. This means that, each year, Jim’s house was worth 100 percent plus 𝑥 percent or 100 and 𝑥 percent of what it was worth the year before. To find a 100 and 𝑥 percent of a number, we can divide it by 100 to find one percent and then multiply by 100 plus 𝑥. This will be equivalent to multiplying by the fraction 100 plus 𝑥 over 100. We can separate this out into the sum of two fractions: 100 over 100 plus 𝑥 over 100. 100 over 100 is just one. So the multiplier simplifies to one plus 𝑥 over 100.

Now we can start to form an equation. The original value of Jim’s house is 300000 pounds. For each of the next three years, the value increases by 𝑥 percent. So our multiplier is one plus 𝑥 over 100. We’re multiplying by this three times. So we have 300000 multiplied by one plus 𝑥 over 100 cubed.

In the fourth year, our multiplier is 1.032. We’re told the value of Jim’s house at the end of the four years. So now we have our equation. 300000 multiplied by one plus 𝑥 over 100 cubed multiplied by 1.032 is equal to 352291.72.

Now we want to solve this equation for 𝑥. So the first step will be to divide both sides of the equation by 300000 and 1.032. This gives one plus 𝑥 over 100 cubed is equal to 352291.72 divided by 300000 multiplied by 1.032. I can evaluate this on my calculator. And it gives 1.137893.

The next step is to cube-root both sides of the equation, giving one plus 𝑥 over 100 is equal to the cube root of 1.137893. And if I evaluate that on my calculator by typing cube root of answer, it gives 1.043999. Now we’re nearly there with solving for 𝑥.

The next step is to subtract one from each side of the equation, giving 𝑥 over 100 is equal to 0.043999. Finally, we need to multiply both sides of the equation by 100, giving 𝑥 is equal to 4.3999990.

Now we haven’t been asked to give our answer to a particular degree of accuracy. But as the percentage increase in the fourth year was given to one decimal place, it makes sense to use the same here. The value of 𝑥 rounded to one decimal place is 4.4. The percentage increase in the market value of all houses during each of the first three years is 4.4 percent.