Concepts: Fractions in Disguise (part 1)

1.1 A percentage is a fraction with denominator 100

The symbol % is read as per cent, from the Latin phrase per centum, which means by the hundred. So 25% means 25 out of every 100: 25 marks in a test marked out of 100, ₹25 from a purse of ₹100, or 25 learners in a class of 100. Once this is clear, most percentage questions become questions about fractions.

A percentage compares a part with a whole cut into exactly 100 equal pieces. Because the denominator is always 100, two percentages sit on the same scale and can be compared straight away. Early in the chapter, two biscuit varieties have sugar content 9/34 and 13/45 of the total. Those fractions are awkward to compare, but written as 26.47% and 28.88% the comparison takes one glance.

The fixed denominator also links percentages to decimals, since 100 is a power of 10. For example 31% = 31/100 = 0.31, and converting between fractions, decimals and percentages stays quick.

1.2 Writing a fraction as a percentage

Any fraction can be rewritten with 100 in the denominator. The chapter shows three routes, and all three agree.

1. Scale the fraction: multiply the numerator and the denominator by the same number so that the denominator becomes 100.
2. Multiply by 100: the fraction times 100 is the percentage value.
3. Set up a proportion: write the fraction equal to x/100 and solve for x.

Worked example 1 (Surya's paint). Surya mixes red and yellow paint to match the sunset. Red makes up 3/4 of the mixture. What percentage of the mixture is red?

Step 1. The fraction is 3 parts out of 4. Step 2. Multiply both numbers by 25, because 4 × 25 = 100. Step 3. The result is 75 out of 100, so red is 75% of the mixture. The remaining yellow is 25%.

Worked example 2 (prize money). Surya saves 2/5 of his prize money to buy a canvas. Write the saving as a percentage.

Step 1. Multiply both numbers of 2/5 by 20, because 5 × 20 = 100. Step 2. 2/5 = 40/100 = 40%. A second route: 2/5 of 100 is 100 divided by 5, then multiplied by 2, which is 40.

Worked example 3 (percentage to fraction). A snack is 24% badam powder by weight. Write 24% as a fraction in lowest terms.

Step 1. 24% means 24 out of 100, so start with 24/100. Step 2. Divide the top and bottom by 2: 12/50. Step 3. Divide both by 2 again: 6/25. The numbers 6 and 25 share no common factor, so the answer is 6/25. In general, z% is any fraction equivalent to z/100.

1.3 Finding a percentage of a quantity

Madhu and Madhav ate biscuits of different varieties. Madhu's variety had 25% sugar and Madhav's had 35% sugar, so the percentages alone do not tell us who ate more sugar. The weight eaten matters as well.

Madhu ate 120 g of biscuits. Sugar is 25 parts in every 100 parts, so the sugar in 120 g is found from a proportion.

Step 1. Write the proportion 25/100 = s/120, where s grams is the sugar. Step 2. Cross multiply: 100s = 25 × 120 = 3000. Step 3. Divide by 100: s = 30. Madhu ate 30 g of sugar. A quick check by ratios: 25 g per 100 g, so 5 g per 20 g, and therefore 30 g per 120 g.

Madhav ate 95 g of biscuits at 35% sugar. The same proportion gives 35/100 = s/95, so s = 35 × 95 / 100 = 33.25 g. Madhav ate more sugar even though he ate fewer biscuits.

More generally, y% of a quantity a is obtained by multiplying a by y/100.

Worked example 4 (Zubin's test). A test is marked out of 75. A student needs 80% or more to get an A grade. What is the lowest score that earns an A?

Step 1. 80% of 75 is 80/100 × 75. Step 2. Simplify 80/100 to 4/5 and multiply: 4/5 × 75 = 60. Step 3. Check with decimals: 0.8 × 75 = 60. Zubin needs at least 60 marks.

Worked example 5 (the cyclist). A cyclist has covered 40% of the Delhi to Agra trip, which is 92 km. How much distance is left?

Step 1. 40% of the total distance is 92 km. Step 2. 20% is half of that, namely 46 km. Step 3. The remaining 60% is three blocks of 20%, so 3 × 46 = 138 km. Step 4. Check with an equation: if the total is T, then 40/100 × T = 92, so T = 230 km, and 230 - 92 = 138 km remain.

1.4 Percentages greater than 100

A percentage can be more than 100. Kishanlal runs a garment shop with a daily target of ₹5000. On the first day he sold goods worth ₹2000, so he reached 2000/5000 × 100 = 40% of the target, and he was 60% short. On the fourth day he sold ₹6000, which is 6000/5000 × 100 = 120% of the target, that is 20% more than he needed. On Day 7 he reached 150% of the target, so his sales were 1.5 × 5000 = ₹7500. A farmer whose harvest rose from 260 kg to 650 kg now has 2.5 times the earlier crop, because 650/260 = 2.5.

Below 100% the part is smaller than the whole. Above 100% the part is larger than the whole, and the percentage tells us how many times the whole it is.

1.5 Mental percentages and the FDP trio

Percentages of round numbers can be computed without pen and paper. First find 10%, then build the rest: 20% is double 10%, 5% is half of 10%, 15% is 10% plus 5%, and 75% is three quarters. Adding and subtracting these blocks keeps the arithmetic simple.

Fractions, decimals and percentages carry the same value in three forms: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/10 = 0.1 = 10%, and 1/100 = 0.01 = 1%. Moving from a percentage to a decimal is a shift of the decimal point two places, and moving back shifts it two places the other way.