Why are percentages taken out of 100?

Any denominator could be used, but 100 suits our base-ten number system: 31% is simply 0.31, so changing between fractions, decimals and percentages is quick. It is also a round number that is easy to picture, and large enough to show detail.

How do you convert a percentage into a fraction?

Write the number over 100 and simplify. For example, 24% = 24/100 = 6/25, and 120% = 120/100 = 6/5. Any fraction equal to 24/100, such as 12/50 or 48/200, also names 24%.

Can a percentage be more than 100?

Yes. A percentage above 100 means more than the whole. If a shop’s target is ₹5000 and it sells goods worth ₹6000, it has reached 120% of the target, which is 1.2 times the target or 20% more than it.

Percentages: Fractions Out of 100

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What does per cent mean?

The sign % is read as ‘per cent’, from the Latin per centum, meaning ‘by the hundred’. So 25% means 25 out of every 100, whether we are counting people, rupees or marks. That makes a percentage just a fraction whose denominator is 100. For example, 20% = 20/100 = 1/5, and 50% of a quantity s is 50/100 × s, which is half of s.

Per cent means out of 100: 25 of 100 squares shaded, so 25% = 25/100 = 1/4 = 0.25 — Per cent means out of 100

From a fraction to a percentage

Surya mixes red and yellow paint, and red makes up 3/4 of the mixture. What percentage is red? One way is to build equivalent fractions until the denominator is 100: 3/4 = 15/20 = 75/100, so red is 75% and yellow is the other 25%. A quicker way is to multiply the fraction by 100: 3/4 × 100 = 75. This works because a fraction tells us a share of one whole, and a percentage tells us the share of 100. So 3/5 = 60%, 9/20 = 45% and 1/3 = 33⅓%. Going the other way, write the percentage over 100: 24% = 24/100 = 6/25.

Turning 3/4 into a percentage with a bar model, equivalent fractions and multiplying by 100 — Turning 3/4 into a percentage

Why bother with percentages?

A biscuit factory makes two new varieties: sugar is 9/34 of the first and 13/45 of the second. Which is sweeter? It is hard to say at a glance. Written as 26.47% and about 28.89%, the answer is obvious: the second. When everything is out of 100, comparing is easy. The idea has a long history: Kautilya’s Arthashastra, from the 4th century BCE, gave fair interest rates ‘per month per cent’, and Italian traders of the 15th century wrote ‘xx p cento’ for 20%.

The FDP trio

Every percentage has a fraction and a decimal that mean the same thing: 50% = 1/2 = 0.5, 25% = 1/4 = 0.25, 10% = 1/10 = 0.1, 1% = 1/100 = 0.01 and 43% = 43/100 = 0.43. Any member of the trio can do the job. To find 50% of 24, multiply by 1/2 or by 0.5: both give 12. To find 80% of 75, work out 4/5 × 75 or 0.8 × 75; both give 60, the minimum mark for an A grade in a test out of 75 where the cut-off is 80%.

Percentage of a quantity in your head

In general, y% of z = y/100 × z. With a few tricks, most of these can be done mentally. To find 10%, divide by 10. Halve that for 5%, double it for 20%, and take a quarter of the number for 25%. Combine them for others: 15% = 10% + 5%, 40% = 2 × 20%, 75% = 50% + 25%, 90% = 100% − 10%, and 55% = 50% + 5%. So 15% of 80 is 8 + 4 = 12, and 75% of 200 is 100 + 50 = 150. Estimate first: since 2/9 lies between 20% and 25%, a calculated answer of 22.22% looks right.

Finding the whole

Sometimes the part is known and the whole is not. A cyclist has covered 92 km, which is 40% of the way. Halving gives 20% = 46 km, so the remaining 60% is 138 km. Or set up 40/100 = 92/d, which gives a whole trip of d = 230 km. Percentages can also be more than 100: if Kishanlal’s target is ₹5000 and he sells ₹7800, he has reached 156% of the target. Read more in the Chapter 1 notes, and see the NCERT solutions for the exercises. The next topic is percentage increase and decrease.

What is a percentage?