### 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.