Equivalent fractions and percentages — Mathematics, 11–13
Two fractions can name the same amount, and a percentage is just a fraction with denominator 100.
Same amount, different names
Two fractions are equivalent when they show the same part of the whole, even with different numbers. To go from one to the other, multiply the numerator and the denominator by the same number: 2/3 = 10/15, because 2 × 5 = 10 and 3 × 5 = 15. A percentage is a fraction with denominator 100: 25% means 25/100, which is equivalent to 1/4.
Why the denominator 100 is handy
Fractions with different denominators are hard to compare at a glance, but once they share a denominator you only compare numerators. The number 100 is one everybody knows: saying 75% is saying “75 parts in every 100”. That is why percentages show up on battery icons, in charts and in the news: one shared way to say “how much of the whole”.
From 3/5 to a percentage
What percentage is 3/5? Step 1: you want an equivalent fraction with denominator 100. Step 2: 100 ÷ 5 = 20, so multiply by 20. Step 3: 3 × 20 = 60 and 5 × 20 = 100, so 3/5 = 60/100. Step 4: 60/100 is 60% and, as a decimal, 0.60. Backwards: 75% = 75/100 = 3/4, because 75 ÷ 25 = 3 and 100 ÷ 25 = 4.
Adding is not the same as multiplying
To find a fraction equivalent to 2/3, many people add the same number on top and on the bottom: 2 + 4 = 6 and 3 + 4 = 7, so they write 6/7. It seems fair, because you did the same thing to both. But 6/7 is about 0.86 and 2/3 is about 0.67: they are not equal. Only multiplying (or dividing) by the same number keeps the part of the whole: 2 × 5 = 10 and 3 × 5 = 15, and 10/15 is the one equivalent to 2/3.
In recipes and on your phone
You see equivalent fractions and percentages every day. In a recipe, 2/4 of a cup is the same as 1/2 a cup. On a phone's battery icon, 75% means 75/100, that is, 3/4 of the bar full; 50% is one half and 25% is 1/4. In charts and in the news, “percentage” always means “so many parts in every 100”.
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