8.3 Percentage of a Quantity

To find y\% of a value, multiply that value by \dfrac{y}{100}:

y\% \text{ of } V = \frac{y}{100}\times V.

Worked Example

Tara ate 140 g of a granola bar that is 30% nuts. How many grams of nuts did she eat?

Set up the proportion 30:100 :: w:140, i.e. \dfrac{30}{100}=\dfrac{w}{140}, so

w = \frac{30}{100}\times 140 = 42\ \text{g of nuts.}

Rohan ate 110 g of a different bar that is 45% nuts. Here \dfrac{45}{100}=\dfrac{w}{110} gives w = \frac{45}{100}\times 110 = 49.5\ \text{g.} So Rohan took in more nuts (49.5 g against 42 g), even though his bar was lighter.

Percentages you can do in your head

Some percentages are friendly precisely because they are tidy fractions:

25\% = \tfrac{1}{4}, \qquad 50\% = \tfrac{1}{2}, \qquad 10\% = \tfrac{1}{10}, \qquad 20\% = \tfrac{1}{5}.

So 25\% of 40 is simply \tfrac14 \times 40 = 10, no pen required.

Math Talk: building blocks Notice that 20\% of a value is twice 10\% of it, and 5\% is half of 10\%. So you can assemble awkward percentages out of easy ones: 15\% = 10\% + 5\%, \qquad 25\% = 20\% + 5\%, \qquad 90\% = 100\% - 10\%. We can even check (20\% \text{ of } y) + (5\% \text{ of } y) = 25\% \text{ of } y symbolically: \tfrac{20}{100}y + \tfrac{5}{100}y = \tfrac{25}{100}y. How would you pin down 70\% or 55\% mentally?

A Mental-Math Shortcut: the 10% anchor Almost any everyday percentage can be reached from 10%, which you find just by moving the decimal point one place left. Want 15\% of a ₹640 bill (say, a tip)? Take 10\% of 640, which is 64; halve it to get 5\%, which is 32; add them: 64 + 32 = \mathbf{96}. To check, 0.15 \times 640 = 96, spot on. Once you can land 10% instantly, 5%, 15%, 20%, 30% and 40% are all a hop or two away.

The FDP Trio: Fractions, Decimals, Percentages

The very same proportion can wear all three costumes at once. Since 50\% = \tfrac{50}{100} = \tfrac12 = 0.5, finding 50\% of 24 is identical whether you compute \tfrac12 \times 24 or 0.5 \times 24, both land on 12.

Per cent 50\% 25\% 75\% 10\% 1\% 5\% 43\%
Fraction \tfrac{1}{2} \tfrac{1}{4} \tfrac{3}{4} \tfrac{1}{10} \tfrac{1}{100} \tfrac{1}{20} \tfrac{43}{100}
Decimal 0.5 0.25 0.75 0.1 0.01 0.05 0.43

Worked Example

A quiz is marked out of 60. An A grade needs 85%. What is the lowest A-grade score?

Three routes, one destination:

  • Fraction: \dfrac{85}{100}\times 60 = \dfrac{17}{20}\times 60 = 51.
  • Decimal: 0.85 \times 60 = 51.
  • Proportion: \dfrac{85}{100}=\dfrac{m}{60}\Rightarrow m = 51.

So a student must score at least 51 marks.

Ratios become percentages

A ratio first becomes a fraction, and then a percentage.

Worked Example

A cold coffee is mixed with coffee decoction and milk in the ratio 3:5. What percentage of the drink is decoction? In 400 ml of the drink, how much decoction is there?

The whole drink is 3 + 5 = 8 parts, so decoction is \tfrac{3}{8} of it:

\frac{3}{8}\times 100 = 37.5\%\ \text{decoction}, \qquad \text{milk} = 100 - 37.5 = 62.5\%.

In 400 ml: \dfrac{37.5}{100}\times 400 = \mathbf{150\ ml} of decoction.

Estimate First Before grinding out \tfrac{3}{8}, notice \tfrac{3}{8} is a touch under half (half of 8 is 4) and well above a quarter (a quarter of 8 is 2). And since 10\% of 8 is 0.8 while 50\% is 4, the answer ought to settle somewhere near 40%, which 37.5\% does. Estimating before you calculate sharpens your number sense and traps slips before they spread.

Working backwards, and percentages above 100

Sometimes it is the part that is handed to us, and we must rebuild the whole.

Worked Example

A runner has covered 84 km, which is 35% of an ultramarathon route. How many kilometres remain?

If 35\% is 84 km, then the full 100\% is \tfrac{84}{35}\times 100 = 240 km. The remaining 65\% is

240 - 84 = \mathbf{156\ km}.

Can a percentage push past 100? Of course, it merely means more than the whole. If a stall owner’s daily goal is ₹8000 and she rings up ₹9600, she has hit

\frac{9600}{8000}\times 100 = 120\% \text{ of her goal} \;(\text{that is, } 20\% \text{ over}).

A mango harvest of 720 kg set against last year’s 240 kg is \dfrac{720}{240}\times 100 = 300\%, in plain words, three times the earlier crop.

Figure it Out: Percentage of a Quantity

Practice: estimate first!

  1. Find:   (i) 25% of 240   (ii) 16% of 350   (iii) 62% of 450   (iv) 140% of 60   (v) 1% of 1 hour   (vi) 7% of 20 kg.
  2. Kabir made 80 ml of pale blue paint; blue is \tfrac45 of it. How much blue did he use?
  3. Fill in:   (i) 40% of k is 90, so 80% of k = \underline{\ }, 120% of k = \underline{\ }, 160% of k = \underline{\ }.   (ii) 100% of m is 320, so 10% of m = \underline{\ }, 1% of m = \underline{\ }, 6% of m = \underline{\ }.
  4. Fill the blanks:   (i) 4 is \underline{\ }% of 400.   (ii) \underline{\ } is 35% of 4.   (iii) 60 is 75% of \underline{\ }.
  5. Is 10% of a day longer than 1% of a week?
  6. Tea pickers take 18 days to cover 25% of an estate. At the same rate, how long for the whole estate? Why does “same rate” matter?
  7. A 120-minute workshop has intro : main : wrap-up = 15\% : 70\% : 15\%. How long is each part?
  8. A halwa recipe for 4 people uses Rava 40%, Sugar 40%, Ghee 20%.   (i) For 8 people, what are the proportions?   (ii) If the total is 2.5 kg, how much of each?
    1. \tfrac14\times 240 = \mathbf{60}.   (ii) 0.16\times 350 = \mathbf{56}.   (iii) 0.62\times 450 = \mathbf{279}.   (iv) 1.4\times 60 = \mathbf{84}.   (v) 1% of 60 min = \mathbf{0.6\ min} = 36 s.   (vi) 7% of 20 kg = \mathbf{1.4\ kg} = 1400 g.
  2. \tfrac45\times 80 = \mathbf{64\ ml} of blue.
    1. 40\% of k is 90 \Rightarrow k = \tfrac{90}{0.4} = 225; then 80\% = \mathbf{180}, 120\% = \mathbf{270}, 160\% = \mathbf{360}. (Each is a multiple of the 40\% value: 180 = 2\times 90, etc.)   (ii) 10\% = \mathbf{32}, 1\% = \mathbf{3.2}, 6\% = \mathbf{19.2}.
    1. \tfrac{4}{400}\times 100 = \mathbf{1}.   (ii) 0.35\times 4 = \mathbf{1.4}.   (iii) \tfrac{60}{0.75} = \mathbf{80}.
  5. 10% of a day = 0.1\times 24 = 2.4 h; 1% of a week = 0.01\times 7\times 24 = 1.68 h. Yes, 10% of a day is longer.
  6. 25\% \to 18 days, so 100\% \to \tfrac{18}{0.25} = \mathbf{72\ days}. The assumption is needed because the calculation only holds if the picking speed never changes.
  7. Intro = 15\% of 120 = \mathbf{18} min, main = 70\% = \mathbf{84} min, wrap-up = \mathbf{18} min.
    1. The proportions stay the same, 40%, 40%, 20%, only the total doubles.   (ii) Of 2.5 kg: Rava = 0.4\times 2.5 = \mathbf{1\ kg}, Sugar = \mathbf{1\ kg}, Ghee = 0.2\times 2.5 = \mathbf{0.5\ kg}.