If there is a 20 percent discount on a TV set, calculate the percentage by which the sale price must be increased in order to get the original price.
In this question, we’re told that the price of the TV set, which we can call the original TV price, has been discounted by 20 percent to get to a sale price. Which means that if we find 20 percent of the TV price and subtract, this will give us the sale price.
It can be very tempting to think if we add 20 percent to the sale price, we would get back to the original TV price. But this would be incorrect, since the percentage we add to the sale price is a percentage of it and not the original TV price. 20 percent of the smaller sale price value cannot be the same as 20 percent of the larger original TV price.
So let’s think about this in terms of the multipliers that we can use. If we call the original TV price 𝑎 and the sale price 𝑏, then we can write 𝑏 equals 80 percent of 𝑎, since we’ve taken away 20 percent of 𝑎 to get to 𝑏. We can then write our 80 percent as a fraction, giving us 𝑏 equals 80 over 100 times 𝑎. We can then simplify our fraction to give us 𝑏 equals four-fifths times 𝑎. So therefore, to get from our original TV price 𝑎 to our sale price 𝑏, we can write that our multiplier is four-fifths.
So therefore, if we want to get from our sale price back to our original TV price, we can divide by four-fifths. And to express this in terms of a multiplication rather than a division, we can say that this is the same as multiplying by five over four. We can do this because when we divide by a fraction, we change the division to a multiplication and we flip the numerator and denominator.
So now we have worked out that to get from our sale price to our original TV price, we multiplied by five-fourths. But we need to express this as a percentage. To do this, we write our multiplier five-fourths times 100, giving us 500 over four, which we can write as 125 percent. Since this value 125 percent also includes the 100 percent of the sale price, then we must subtract 100 percent to give us 25 percent.
So we can say that the percentage by which the sale price must be increased to get the original price is 25 percent.