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It is common to find in daily life changes in values given by a percentage. A discount on the price of a product, or a surcharge for late payment of an invoice, is usually given as a percentage; in the first case it is a decrease in the original value, while in the second it results in an increase. Both a decrease and a percentage increase involve the application of a factor that leads to the change from the initial value.
The calculation of the percentage change
If you want to calculate the percentage variation when you have both values, the initial one and the result of applying the percentage, you must make the difference between the final and initial values to determine the difference between the two values; then divide the result of the subtraction by the initial value to normalize the variation, and multiply this last value by 100 to obtain the difference expressed as a percentage. If the resulting value is positive, it means that the final value is greater than the initial one, and therefore there is an increase in the initial value. If, on the contrary, the result is a negative number, the final value is less than the initial one and there is a decrease.
Let’s see an example. If the price of a kilo of apples is 3 pesos but on Tuesdays there is a special discount and it is sold at 1.80 pesos, what is the discount on Tuesdays? The original value is 3 and the value returned on Tuesdays is 1.80; the difference between the final value and the initial value is -1.20. If this value is divided by 3 (initial value) the result is -0.40. And multiplying it by 100 results in a variation of -40 percent, which is expressed with the symbol %: -40%. Since the percentage change is negative, we have a decrease from the original price, a discount, and the final result is stated as a 40% discount on the price of apples on Tuesdays.
The calculation of the final value
However, it is more common to find the reverse situation; that is, having the original value and a discount or increase percentage, a situation in which we must find out the final value once the percentage has been applied. In the previous case it is announced that on Tuesdays there is a 40% discount in the price of apples and we need to calculate what would be the final value that we would have to pay knowing that the usual price is 3 pesos. First you have to convert the percentage discount value into a fraction, dividing it by 100. Then we multiply it by the initial value of the apples and we get the discount offered. And finally, if you want to know the price you have to pay, you have to subtract the discount from the original price.
In our example, dividing 40 by 100 gives 0.4; multiplying the discount fraction by the original price, 3 pesos, you get 1.2. That is, the apple vendor offers a discount of 1.2 pesos on Tuesdays. And to obtain the value that we must pay, the discount is subtracted from the original price, and it is obtained that on Tuesdays the value of the kilo of apples is 1.8 pesos.
In the event that an increase in value is proposed, the calculation is similar. For example, we forgot to pay the electricity service bill, which was 5 pesos, so we have to pay it with a surcharge of 10%. First, the percentage value is converted to a fraction by dividing it by 100, and 0.1 is the result. This value is then multiplied by the original value of our electric service bill to find out the value of the surcharge, which in this case would be 0.5 pesos. Unlike the previous example, since it is a surcharge, an increase with respect to the initial value, the value of the surcharge must be added to the initial value. So we will have to pay 5.5 pesos for paying the bill late.
three examples
- If looking for offers we find that a pair of shoes that previously cost 4 pesos is now being offered at 3, what is the percentage value of the discount? The difference between the final value and the initial value is -1; divided by the initial value gives -0.25, and multiplied by 100 we get the value -25. We know that it is a discount, so we change the sign and it is stated that the discount on the pair of shoes is 25%.
- Winter is coming to an end and warm country boots are 40% off. What is the final value of a jacket that cost 8 pesos? First convert the percentage discount to a fraction by dividing by 100; the result is 0.4. Multiplying this value by the original cost of the jacket, the discount is obtained, which in this case is 3.2 pesos. And subtracting the discount from the original value, the final value of the jacket is obtained, 4.8 pesos.
- We need to borrow 100 pesos from the bank, and the interest on the loan is 5%. What is the total amount that we must return to the bank? Converting the interest rate from percentage to fraction, dividing by 100, gives a value of 0.05; multiplying this value by the amount of the loan is obtained is the amount that must be paid to the bank for the service. The product of 100 times 0.05 is 5 pesos. Adding the cost of the loan to the amount borrowed, the total amount that must be paid to the bank is obtained; in this case it is 105 pesos.
Fountain
Samuel Selzer, Algebra and analytical geometry. Second edition. Buenos Aires, 1970.
