Fractions, Ratio and Percent Explained with Animation

Equal parts of a whole

Take a chocolate bar and break it into four pieces of exactly the same size. One piece is one quarter, written 1/4. The bottom number, the denominator, says how many equal parts the whole is cut into. The top number, the numerator, says how many of them you take. The parts must be equal, or the fraction means nothing. Eat 3 of the 8 slices of a pizza and you have eaten 3/8. In the drawing a whole bar is cut and one part lights up. People used fractions very early: the Rhind papyrus, copied in Egypt around 1550 BC, writes most of them as sums of pieces with 1 on top.

One amount, many names

Cut the same bar into 2, 4 or 8 pieces and see how the bars line up. One half is exactly as long as two quarters and as four eighths: 1/2 = 2/4 = 4/8. These are equivalent fractions. The rule is simple: multiply the top and the bottom by the same number and the value does not change. From 1/2, multiplying by 2 gives 2/4, and multiplying by 4 gives 4/8. Why does it work? Because 2/2 and 4/4 are both equal to 1, and multiplying by 1 changes nothing. In the picture, the edges of the bars meet at the same place.

Making a fraction simpler

Read the rule backwards and you can simplify. Divide the top and the bottom by the same number. Take 6/8. Both 6 and 8 divide by 2, so 6/8 = 3/4. Nothing more can be done, because 3 and 4 share no factor except 1, so the fraction is in its simplest form. Another case: 18/24. The biggest number dividing both is 6, so 18/24 = 3/4 as well. Notice that 6/8 and 18/24 are the same amount, three quarters of a bar. Try it in the sandbox under the picture: choose a numerator and a denominator, and the bar, circle, decimal and percent all update together.

Adding with a common denominator

Can you add 1/2 and 1/3 by adding the tops and the bottoms? No: that gives 2/5, which is less than 1/2 alone, so it must be wrong. Pieces of different sizes cannot be counted together. First cut both into the same size. The smallest common size for halves and thirds is sixths: 1/2 = 3/6 and 1/3 = 2/6. Now the pieces match, so we add the tops and keep the bottom: 3/6 + 2/6 = 5/6. Subtraction works the same way: 3/4 − 1/6 = 9/12 − 2/12 = 7/12. As a check with decimals, 0.75 − 0.1667 is about 0.5833, and 7/12 is 0.5833.

Ratio and proportion

A ratio compares two amounts. A pancake recipe asks for 2 eggs to 3 cups of flour, written 2 : 3. To feed a bigger crowd, multiply both numbers by the same factor. Times 4 gives 8 eggs to 12 cups, and the proportion 2/3 = 8/12 stays true. A quick check is to cross-multiply: 2 × 12 = 24 and 3 × 8 = 24. Ratios turn into fractions when you look at the whole. In a mix of 2 parts blue paint to 3 parts white there are 5 parts altogether, so blue is 2/5 of the paint. In 10 litres that means 4 litres of blue and 6 litres of white.

Percent means per hundred

The word percent comes from the Latin for 'by the hundred'. Picture a square grid of 10 by 10 small squares, 100 cells in all. If 25 of them are coloured, then 25 percent of the grid is coloured, written 25 %. That is also 25/100, which simplifies to 1/4, and it is 0.25 as a decimal. Fractions, decimals and percents are three ways of writing one amount. To turn a fraction into a percent, divide the top by the bottom and multiply by 100. For 3/8: 3 ÷ 8 = 0.375, and 0.375 × 100 = 37.5 %. Move the sliders in the sandbox and read all three forms.

Percent of a number

To find a percent of a number, turn the percent into a decimal and multiply. Fifteen percent of 60 is 0.15 × 60 = 9. A shop takes 20 % off an item priced at 80. The discount is 0.20 × 80 = 16, so you pay 80 − 16 = 64. You can also keep 80 % in one step: 0.80 × 80 = 64. Both roads arrive at the same price. A common trap is mixing up what the percent is taken from. Raise a price of 100 by 10 %, then lower it by 10 %, and you get 110, then 99, not 100, because the second 10 % is taken from the larger number.

Fractions, ratio and percent