Fractions in Disguise · Lesson 2 of 12
Percentage of Some Quantity
“Find percentages of quantities using bar models, proportional reasoning and mental calculation.”
• Identify the whole before calculating a percentage. • Find a percentage of a quantity using equal parts, proportions and multiplication. • Calculate useful percentages mentally by combining familiar shares. • Compare actual quantities when percentages refer to different wholes.
Two friends choose different biscuits. One packet contains 25% sugar and the other contains 35% sugar. It is tempting to say immediately that the second friend eats more sugar. But imagine that the first friend eats a large bowl of biscuits while the second eats only one small biscuit. To compare the actual sugar eaten, we need two pieces of information: the sugar proportion and the mass of biscuits each person eats.
A percentage acts on a whole quantity. The words “25% sugar” describe a relationship, whereas “30 g of sugar” describes an amount. Learning to move between a proportion and an amount lets you use percentages in recipes, time plans, shopping and measurement.
We will begin by dividing quantities into easy equal shares. Once that reasoning is clear, the multiplication rule will feel like a short version of something you already understand.
Suppose a packet contains 120 g of biscuits and one quarter of its mass is sugar. Dividing the whole mass into four equal parts gives 120 ÷ 4 = 30 g per part. Since 25% is one quarter, the sugar mass is 30 g.
You could also imagine splitting the mass into one hundred equal parts. Each part is 120 ÷ 100 = 1.2 g. Taking twenty-five of those parts gives 25 × 1.2 = 30 g. The two methods describe the same share at different levels of subdivision.
The base is the whole 120 g packet, not the sugar itself. If you used the sugar mass as the base while trying to find the sugar mass, you would lose the given relationship. Write the whole next to 100% whenever you are unsure.
Problem
Find 25% of 120 g.
- 1.The whole is 120 g, so 100% corresponds to 120 g.
- 2.25% = 25/100 = 1/4. Divide the whole into four equal shares: 120 ÷ 4 = 30.
- 3.Therefore 25% of 120 g is 30 g.
- 4.Check: four shares of 30 g total 120 g. The unit stays grams because we took a fraction of a mass.
A proportional method gives the same result. If 100 g of biscuits contains 25 g of sugar, then 20 g contains 5 g of sugar and 120 g contains 30 g of sugar. Each change multiplies or divides both masses by the same factor.
This scaling assumes that the sugar proportion is uniform in the biscuits we are considering. In a recipe or a mixture problem, such an assumption is usually stated or intended. If the composition changes from one portion to another, one percentage may not describe every portion.
The multiplication rule simply says: divide the whole by one hundred to find 1%, then take p of those shares. It works even when the percentage or whole is not convenient for mental division. Simplifying a fraction before multiplying can make the arithmetic easier.
Check the size of your answer before accepting it. For a positive whole, a percentage between 0% and 100% gives a part between zero and the whole. If 35% of 95 g comes out as 3325 g, you have probably forgotten to divide by one hundred.
Problem
Find 35% of 95 g.
- 1.The whole is 95 g and the requested share is 35/100.
- 2.Calculate 95 × 35 = 3325, then divide by 100: 3325 ÷ 100 = 33.25.
- 3.The required mass is 33.25 g.
- 4.For an estimate, 35% of 100 g is 35 g. Since 95 g is slightly smaller than 100 g, 33.25 g is reasonable.
Problem
Madhu eats 120 g of biscuits containing 25% sugar. Madhav eats 95 g containing 35% sugar. Who eats more sugar, and by how much?
- 1.Calculate each share using its own whole. Madhu eats (25/100) × 120 = 30 g of sugar.
- 2.Madhav eats (35/100) × 95 = 33.25 g of sugar.
- 3.Compare masses with the same unit: 33.25 − 30 = 3.25 g.
- 4.Madhav eats 3.25 g more sugar. The conclusion follows from the calculated masses, not from comparing 35 and 25 alone.
Free-hand Computations
Some percentages are especially useful mental building blocks. Ten per cent is one tenth, so divide by ten. One per cent is one hundredth, so divide by one hundred. Fifty per cent is one half, and twenty-five per cent is one quarter.
Build other percentages from these pieces. Twenty per cent is twice ten per cent. Five per cent is half ten per cent. Fifteen per cent is ten per cent plus five per cent. For a whole of 80, these give 10% = 8, 5% = 4 and 15% = 12.
These additions are valid because every percentage is acting on the same whole. Adding 20% of one amount to 5% of a different amount does not generally give 25% of either amount. The shared base is the reason the shortcut works.
| Percentage | Meaning | Value when the whole is 80 |
|---|---|---|
| 1% | Divide by 100 | 0.8 |
| 5% | Half of 10% | 4 |
| 10% | Divide by 10 | 8 |
| 20% | Twice 10% | 16 |
| 25% | Divide by 4 | 20 |
| 50% | Divide by 2 | 40 |
| 75% | Three quarters | 60 |
Problem
Find 15%, 55% and 90% of 240 without long multiplication.
- 1.Begin with useful pieces: 10% of 240 is 24, 5% is 12 and 50% is 120.
- 2.15% = 10% + 5%, so 15% of 240 = 24 + 12 = 36.
- 3.55% = 50% + 5%, so 55% of 240 = 120 + 12 = 132.
- 4.90% = 100% − 10%, so 90% of 240 = 240 − 24 = 216.
- 5.Check that the answers increase as the percentages increase: 36 < 132 < 216.
Complementary shares are another helpful shortcut. If you know 25% of a value, then 75% is the whole minus that quarter. If you know 30%, then 70% is the whole minus 30%. Think of the completed and remaining parts of one journey.
Keep units visible when the whole is time, length or mass. One per cent of an hour is 0.01 hour, which is 0.6 minute or 36 seconds. Writing only “0.01” leaves the answer incomplete and can conceal a unit conversion error.
Problem
A 90-minute session uses 10% of the time for warming up, 80% for play and 10% for cooling down. Find each duration.
- 1.The whole session, 90 minutes, represents 100%.
- 2.10% of 90 minutes is 9 minutes, so warming up and cooling down each take 9 minutes.
- 3.80% is eight groups of 10%, so play takes 8 × 9 = 72 minutes.
- 4.Check the total: 9 + 72 + 9 = 90 minutes. The three shares account for all the time.
Increasing the size of a recipe does not change the percentage of each ingredient when all ingredients are scaled together. A mixture with 40% rava, 40% sugar and 20% ghee still has those percentages after every ingredient is doubled. The masses double, while the proportions stay the same.
A useful problem-solving routine is to name the whole, select an easy fraction or a multiplication method, calculate with units, and check whether the answer has a sensible size. Try estimating first; an approximate expectation helps you catch an extra zero.
Choose the wholes 50, 80, 200 and 287. Find 10%, 20%, 5% and 25% of each. For 287, your results should be 28.7, 57.4, 14.35 and 71.75. Explain why the 20% and 5% columns add to the 25% column in every row. Then use the same table to find 15% and 40%.
Quiz
Two foods contain 20% and 30% sugar. Which statement must be true?
What is 5% of 360?
Why can 20% of W and 5% of W be added to get 25% of W?
What is 1% of one hour?
A recipe is doubled while its proportions stay the same. What happens to a 20% ingredient?
Practice Problems
- Calculate 25% of 160 and 16% of 250. Solution: 25% is one quarter, giving 160 ÷ 4 = 40. For 16%, calculate (16/100) × 250 = 40. Different percentage-and-whole pairs can produce the same amount.
- Find 7% of 10 kg, expressing the answer in kilograms and grams. Solution: 1% of 10 kg is 0.1 kg. Seven such parts give 0.7 kg. Since 1 kg = 1000 g, this equals 700 g.
- A 2 kg halwa mixture contains 40% rava, 40% sugar and 20% ghee. Find the mass of each. Solution: Rava = 0.4 × 2 = 0.8 kg; sugar is also 0.8 kg; ghee = 0.2 × 2 = 0.4 kg. The sum 0.8 + 0.8 + 0.4 = 2 kg checks the calculation.
- Is 10% of a day longer than 1% of a week? Use one common unit. Solution: One day is 24 hours, so 10% is 2.4 hours. A week is 168 hours, so 1% is 1.68 hours. Thus 10% of a day is longer by 0.72 hour, or 43.2 minutes.
- A person eats 200 g of a food containing 15% sugar; another eats 80 g of a food containing 30% sugar. Compare their sugar intake. Solution: First intake = 0.15 × 200 = 30 g. Second intake = 0.30 × 80 = 24 g. The first person eats 6 g more despite choosing the food with the smaller percentage, because their whole portion is larger.
Key Takeaways
• A percentage gives a proportion; the whole determines the actual amount. • Calculate p% of W by multiplying W by p/100. • Use familiar fractions and 1%, 5% or 10% as mental building blocks. • Combine percentage shares directly only when they use the same whole. • Keep units attached to calculated amounts and convert them carefully. • Scaling all parts and the whole equally preserves the percentages.