Percentages
A sale in a Dublin shop: 50% off plus an extra 20% off. Is that a 70% discount? A price with 23% VAT already included. How can you tell how much it cost without VAT? Percentages are everywhere, and mastering them stops you from being fooled by an advert or an invoice.
Revision: calculating a percentage
One quantity as a percentage of another
To write $a$ as a percentage of $b$, divide and multiply by $100$: $\dfrac{a}{b} \times 100\%$. The part goes on top and the whole goes on the bottom.
Of the $30$ students in a class, $18$ are girls. What percentage of the class are girls? And boys?
Of the $30$ students in a class, $18$ are girls. What percentage of the class are girls? And boys?
What percentage does $45$ represent out of $180$?
What percentage does $45$ represent out of $180$?
The price of a comic has gone from $\text{€}\ 12$ to $\text{€}\ 15$. What percentage has it gone up by?
The price of a comic has gone from $\text{€}\ 12$ to $\text{€}\ 15$. What percentage has it gone up by?
Calculate mentally: a) $25\%$ of $240$. b) $10\%$ of $380$. c) $50\%$ of $96$. d) $75\%$ of $800$.
Calculate mentally:
a) $25\%$ of $240$.
b) $10\%$ of $380$.
c) $50\%$ of $96$.
d) $75\%$ of $800$.
In a village of $2\,500$ inhabitants, $8\%$ are under $15$ years old. How many people under $15$ years old live in the village?
In a village of $2\,500$ inhabitants, $8\%$ are under $15$ years old. How many people under $15$ years old live in the village?
Increases, discounts and chained percentages
Simple interest is different: the interest is always $3\%$ of the original amount, so it is $2000 \times 0.03 = €60$ every year. Simple interest $= \text{principal} \times \dfrac{\text{rate}}{100} \times \text{years}$. Compound interest is a chain of increases, one per year.
A pair of trainers costing $\text{€}\ 60$ is reduced by $15\%$. How many euros do you save? How much do you pay?
A pair of trainers costing $\text{€}\ 60$ is reduced by $15\%$. How many euros do you save? How much do you pay?
A comic shop offers the second item at $50\%$ of its price. If a volume of manga costs $\text{€}\ 9.95$: a) How much do two volumes cost? b) What total percentage…
A comic shop offers the second item at $50\%$ of its price. If a volume of manga costs $\text{€}\ 9.95$:
a) How much do two volumes cost?
b) What total percentage discount is applied on the purchase of the two volumes?
A sports shop advertises: "Sale: $50\%$ off plus an extra $20\%$ off: a total discount of $70\%$!". A pair of trainers at the shop costs $\text{€}\ 120$. Explain the…
A sports shop advertises: "Sale: $50\%$ off plus an extra $20\%$ off: a total discount of $70\%$!". A pair of trainers at the shop costs $\text{€}\ 120$. Explain the error in the advert, calculate the actual percentage discount applied and the correct final price of the trainers.
Undoing a percentage: VAT and the base price
When you know the final price with a percentage already included, to undo it you divide by the multiplier instead of multiplying.
I paid $\text{€}\ 59.50$ for a pair of trousers discounted by $15\%$. What was the price before the discount?
I paid $\text{€}\ 59.50$ for a pair of trousers discounted by $15\%$. What was the price before the discount?
Aoife bought a pair of headphones on a website for €96.80. This price already included a discount of $20\%$ on the base price and a $21\%$ VAT rate applied after the…
Aoife bought a pair of headphones on a website for €96.80. This price already included a discount of $20\%$ on the base price and a $21\%$ VAT rate applied after the discount. What was the base price of the headphones, without discount or VAT?
Recovering a decrease: percentages are not symmetrical
A video game that cost $\text{€}\ 60$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%$. a) What was the price…
A video game that cost $\text{€}\ 60$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%$.
a) What was the price after the increase?
b) What is the price after the drop?
c) Has it returned to the original price? If not, what is the total percentage change?
The ticket price for a water park dropped by $20\%$ at the end of the summer. When the next season arrives, the park wants to restore the exact original price. By what…
The ticket price for a water park dropped by $20\%$ at the end of the summer. When the next season arrives, the park wants to restore the exact original price. By what percentage must the reduced price increase? Explain why the answer is not $20\%$.
A bookshop's sales fell by $35\%$ in February compared to January. In March, sales recovered to reach the January level again. By what percentage did March sales…
A bookshop's sales fell by $35\%$ in February compared to January. In March, sales recovered to reach the January level again. By what percentage did March sales increase compared to February? Round your answer to two decimal places.
Back to the start: A discount of 50 % plus an extra 20 % is equivalent to paying $0.50 \cdot 0.80 = 0.40$ (a real reduction of 60 %, not 70 %); and to find the price without VAT, divide the final price by $1.23$.