Fractions in Disguise · Lesson 12 of 12
Chapter Summary and Practice
“Revise every major chapter concept and formula, interpret diagrams and solve mixed problems.”
• Connect fractions, decimals, ratios and percentages in one coherent method. • Review direct percentages, reverse percentages and percentage changes. • Choose the correct base in shopping, profit, interest and growth problems. • Interpret diagrams and percentage data without confusing proportions with counts. • Solve mixed problems and check calculations, units and practical meaning.
A percentage question can look like a paint recipe, a shopping offer, a growing population or a savings calculation. Underneath these stories, one relationship keeps returning: an amount is being compared with a reference quantity. Once you identify what counts as one hundred per cent, you can decide whether to find a part, recover a whole, measure a change or apply a sequence of changes. This revision brings those decisions together so you can recover the ideas after a break.
Use the examples as opportunities to explain the method aloud. Before calculating, name the base and estimate the likely size of the answer. Afterward, check whether the answer describes an amount, a percentage, a final value or a change. Many errors are errors of interpretation rather than multiplication.
The review follows the same path as your learning: equivalent forms first, then quantities and comparisons, then buying and selling, and finally growth and more demanding percentage reasoning.
Fractions as Percentages
A percentage is a fraction expressed relative to one hundred. The symbol % means “per hundred”, so p% = p/100. A fraction a/b becomes a percentage by calculating 100a/b as the percentage number. For example, 3/5 = 60/100 = 60%.
To reverse the conversion, write the percentage number over one hundred and simplify. Thus 24% = 24/100 = 6/25. The numerator and denominator must be scaled together when you form equivalent fractions. Percentages may contain decimals or fractions: 1/3 = 33⅓% exactly, while 33.33% is a rounded approximation.
Percentage of Some Quantity
A percentage alone describes a proportion. To obtain an actual amount, apply it to the whole. If a 120 g packet is 25% sugar by mass, the sugar mass is one quarter of 120 g, or 30 g. A different packet with the same percentage but a different mass contains a different sugar mass.
Useful mental shares include 50% = one half, 25% = one quarter, 20% = one fifth, 10% = one tenth and 1% = one hundredth. Build 15% as 10% + 5%, or 90% as 100% − 10%. Those additions and subtractions must use the same whole.
The FDP Trio — Fractions, Decimals, and Percentages
A fraction, decimal and percentage can be equivalent names for one value. For example, 3/8 = 0.375 = 37.5%. Divide a percentage number by one hundred to obtain a decimal multiplier; multiply a decimal by one hundred to obtain its percentage number.
Choose the form that makes the arithmetic transparent. Five eighths of 64 is easy to calculate as 64 ÷ 8 × 5 = 40. The equivalent 62.5% or 0.625 describes the same calculation. A quick comparison with familiar benchmarks helps you catch decimal errors.
For a mixture ratio such as 2:7, total parts are 9. The first ingredient occupies 2/9 of the mixture, not 2/7. The denominator must include every ingredient in the whole. Use exact fractions through a calculation and round only the final result when appropriate.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/8 | 0.125 | 12.5% |
| 1/4 | 0.25 | 25% |
| 3/8 | 0.375 | 37.5% |
| 1/2 | 0.5 | 50% |
| 3/4 | 0.75 | 75% |
| 5/4 | 1.25 | 125% |
Finding the Whole and Percentages Greater than 100
To recover a whole, undo the direct-percentage multiplication. If 40% of a journey is 92 km, the total is 92 ÷ 0.4 = 230 km. The remainder is 60% of 230 km, or 138 km. A bar divided into five equal 20% portions shows why the calculation works.
A percentage can exceed one hundred when the compared amount exceeds its reference. Sales of ₹6000 against a ₹5000 target equal 120% of that target. The sales are 20% above target. Likewise, a harvest that is 250% of the earlier harvest represents a 150% increase, because the original quantity already accounts for 100%.
Using Percentages to Compare Proportions
Compare each part with its own whole. A survival count of 72 out of 80 is 90%, whereas 190 out of 250 is 76%. The first group has the higher survival proportion, but the second has more surviving plants. Counts and proportions answer different questions.
For ingredient labels, use one measurement basis, such as mass. A complete set of non-overlapping exact ingredient shares totals 100%. Rounded displays may differ slightly. If a 65 g packet contains 70% potato, 24% oil, 3% salt and 3% spice, their masses are 45.5 g, 15.6 g, 1.95 g and 1.95 g. These total 65 g.
A percentage graph often reports separate bases for separate groups. You cannot add two groups’ percentages to obtain their combined percentage without considering their sizes. Nor can a higher percentage tell you that the group itself contains more people.
Percentage Increase or Decrease
A percentage change measures the difference relative to the original quantity. A rise from ₹30 to ₹42 is an increase of ₹12, which is 40% of ₹30. The new price is 140% of the old price. A fall from 160 to 100 removes 60, which is 37.5% of 160.
A p% increase uses a multiplier of 1 + p/100; a p% decrease uses 1 − p/100. To recover the original value, divide the final value by that multiplier. If an amount becomes 90 after a 20% increase, its original value is 90 ÷ 1.2 = 75. Subtracting 20% of 90 would apply the percentage to the wrong base.
Profit and Loss
Cost price is what a seller paid for an item; marked price is the price displayed before discount; selling price is the actual sale amount. Profit is selling price minus cost when the selling price is larger. Loss is cost minus selling price when the cost is larger.
For standard profit or loss percentages on cost, divide the profit or loss by cost price. A ₹400 item sold for ₹500 makes ₹100 profit, or 25% on cost. A ₹600 item sold for ₹540 makes ₹60 loss, or 10% on cost. Combining the two gives ₹40 profit on ₹1000 total cost, or 4%; averaging their percentages would be wrong.
Recover cost by recognising what percentage of cost the selling price represents. A ₹50 sale at 25% profit means ₹50 is 125% of cost, so cost is ₹40. A 12% loss means that selling price is 88% of cost.
Discounts and Taxes
A discount reduces the price to which it applies. For one discount of d%, the customer pays (100 − d)% of marked price. A separately added tax increases a stated taxable amount. A 10% discount on ₹2000 leaves ₹1800; a stated 5% tax on that discounted amount adds ₹90, giving ₹1890.
Discount percentage uses the quoted price as its base, while profit percentage on cost uses cost. A seller can offer a discount and still profit. Tax collected separately for the government should not be treated as extra profit in these models.
Revenue is the sales amount, excluding separately identified tax. Gross profit is revenue minus cost of goods sold. Net profit is gross profit minus separately listed operating expenses. A profit percentage of revenue differs from a percentage on cost because its denominator differs. Always state which base is intended.
Growth and Compounding
Principal is the amount used as the interest base. Interest is the extra money earned or charged. Amount combines principal with the interest included in the calculation. A per-annum rate is a rate per year; the time and rate periods must match.
With simple interest, the original principal stays the base, so equal periods add equal interest amounts. With compounding, each period’s interest is added back and becomes part of the next period’s base. At 10% p.a., ₹6000 produces ₹7800 in total after three years without compounding, or a balance of ₹7986 with annual compounding.
Using decimal rate r, simple interest is Prt, simple total amount is P(1 + rt), and compound amount is P(1 + r)ᵗ. Compound interest is the final amount minus P. When R is a percentage number instead, substitute R/100 for r. For example, use r = 0.10 or R = 10 for ten per cent, never r = 10 in the decimal-rate formula.
Decline and Repeated Percentage Change
Depreciation is a reduction in value over time. A constant percentage decline model retains the same fraction of the current amount each period. Two successive 5% declines use 0.95 × 0.95 = 0.9025, leaving 90.25% of the original and giving a total 9.75% decrease.
For varying changes, multiply each period’s factor. Successive 3% and 4% rises use 1.03 × 1.04 = 1.0712, an overall 7.12% rise. Equal opposite percentages do not usually cancel because their bases differ. A 10% rise followed by a 10% fall leaves 99% of the original.
Preserve intermediate precision. A model population can have a decimal intermediate value even though actual people are counted in whole numbers. Round the final estimate to an appropriate whole count and state the precision. For exact-count data, a fractional result may reveal inconsistent assumptions instead.
Tricky Percentages
A gain of ₹200 on ₹100 is a larger percentage gain than ₹500 on ₹1000, yet ₹500 is the larger absolute gain. A useful decision needs both the percentage and its base. The same issue arises when comparing a fixed discount with a percentage discount.
Successive discounts multiply the shares retained: 30% off followed by 20% off retains 0.7 × 0.8 = 0.56, giving a 44% discount. A 50% markup followed by 50% off retains 1.5 × 0.5 = 0.75 of cost, a 25% loss. To reverse the markup exactly requires a 33⅓% discount on the marked price.
A percentage of a subgroup uses multiplication of shares. If 40% of a group belongs to a class and 60% of that class subgroup are girls, the share of the whole is 0.4 × 0.6 = 0.24, or 24%. Reversing a comparison instead requires a reciprocal: if A = 1.2B, then B = A/1.2, or 83⅓% of A.
For identical items in “buy two, get one free”, the free share is one of the three items received, so the discount is 33⅓%. Keep practical bundle conditions visible. For numerical calculations, x% of y equals y% of x because both equal xy/100; this is a multiplication identity, not a rule that percentage comparisons can always be reversed unchanged.
| Question asked | Base or method to use |
|---|---|
| What is p% of W? | Multiply W by p/100. |
| A equals p% of what? | Divide A by p/100. |
| How large was the change? | Compare the difference with the original. |
| Profit percentage on cost? | Divide profit by cost price. |
| Discount percentage? | Compare reduction with the price before that discount. |
| Successive percentage changes? | Multiply the final-value factors. |
| Which group has more people? | Percentages alone may be insufficient; examine group totals. |
Figure it Out
Area diagrams provide another way to reason about percentages. Equal side lengths and a diagonal can show a share without measuring every small region separately. In the diagram, the complete square is divided into a left half and a right half. The lower-left quarter is then divided diagonally into two equal triangles.
Region E is one of those two triangles, so it is half of one quarter: 1/2 × 1/4 = 1/8 of the whole square, or 12.5%. The diagonal divides a rectangle into equal-area triangles because both share equal base-and-height area relationships.
A percentage bar chart must also be read with attention to its bases. The following selected values describe the ability to use a computer in four age groups in a 2023 survey. Each female percentage is relative to the female population in that age group; each male percentage is relative to the male population in that age group.
Within both displayed categories, the twenties group has a higher percentage than the teenage group. The chart does not tell you the total number of teenagers or adults in their twenties. A longer percentage bar therefore cannot establish which group contains more people.
Problem
In the chart, 26% of females and 37% of males in their twenties can use a computer. Does this mean 63% of all people in their twenties can do so?
- 1.No. The two percentages use separate group bases. Adding 26 and 37 counts proportions of different wholes.
- 2.To illustrate, imagine 100 females and 100 males with those exact rates. The able counts would be 26 and 37, giving 63 out of 200 people.
- 3.That combined share would be 31.5%, not 63%. This numerical result assumes equal group sizes only.
- 4.Without the group sizes, the exact combined percentage is unknown. It lies between 26% and 37% if both groups are present, so it cannot be 50% or 63%.
- 5.The seniors’ two displayed rates, 2% and 4%, are both below 10%, supporting the statement that fewer than one in ten in that displayed age group can use a computer.
For the thirties group, the displayed percentages are 14% and 25%. A combined percentage cannot exceed both subgroup percentages. The graph therefore does not support a claim that more than one quarter of all people in their thirties can use a computer.
Separate survey percentages describe the proportions observed; they do not explain causes, predict any individual’s ability or reveal the population sizes. Keep the conclusion as specific as the evidence. This is the same base-checking habit used for ingredient labels and sales comparisons.
Problem
A mixture uses red and yellow paint in the ratio 3:2. If it contains 180 ml red paint, find the total volume and yellow percentage under the additive-volume model.
- 1.Total ratio parts = 3 + 2 = 5. Red occupies 3/5 = 60%; yellow occupies 2/5 = 40%.
- 2.The 180 ml red portion is 60% of the whole.
- 3.Whole volume = 180 ÷ 0.60 = 300 ml.
- 4.Yellow volume = 300 − 180 = 120 ml, and 120/300 × 100 = 40%.
- 5.The ratio 180:120 simplifies to 3:2, checking both the reverse calculation and the mixture interpretation.
Problem
The selling price of three identical pencils equals the cost price of five. Find the profit percentage on cost.
- 1.Let the cost of one pencil be c and the selling price of one be s.
- 2.The statement gives 3s = 5c, so s = (5/3)c.
- 3.Profit per pencil = s − c = (5/3 − 1)c = (2/3)c.
- 4.Profit percentage on cost = [(2/3)c ÷ c] × 100 = 66⅔%.
- 5.For a check, choose cost ₹3 per pencil. Three pencils cost ₹9 and sell for ₹15, giving ₹6 profit, or 6/9 × 100 = 66⅔%.
Problem
A seller buys an item for ₹800, sets its marked price at ₹1250 and gives a 20% discount. A stated 5% tax is charged on the discounted price. Find the final bill and profit percentage on cost before other expenses.
- 1.Discounted selling price = 1250 × 0.80 = ₹1000.
- 2.Tax on that base = 1000 × 0.05 = ₹50. Customer payment = ₹1050.
- 3.Seller’s profit before other expenses = ₹1000 − ₹800 = ₹200.
- 4.Profit percentage on cost = 200/800 × 100 = 25%.
- 5.The marked-price discount, cost-based profit and tax calculation use three clearly identified roles; the tax is not counted as extra profit.
Problem
A model quantity begins at 500, grows by 10% per period for two periods, then decreases by 20%. Find the final quantity and its percentage change from the start.
- 1.Two growth periods give 500 × 1.10² = 605.
- 2.The 20% decline retains 80% of 605: 605 × 0.80 = 484.
- 3.Compared with the original 500, the decrease is 16.
- 4.Overall decrease percentage = 16/500 × 100 = 3.2%.
- 5.The complete multiplier 1.10² × 0.80 = 0.968 confirms that 96.8% of the original remains.
Peaceful Knights
For an optional reasoning puzzle, place eight knights on a chessboard so that no knight can attack another. A knight moves two squares in one direction and then one square at right angles. This movement always changes the colour of the square on which it lands.
A simple solution is to put the eight knights on eight distinct squares of the same colour. Every possible destination of a knight lies on the opposite colour, so none of the other seven knights can be attacked. For example, using standard coordinates with a1 dark, a1, c1, e1, g1, b2, d2, f2 and h2 are all dark squares. This puzzle is an optional exercise in recognising a useful invariant.
Quiz
Which calculation recovers a total when 35% of it is 84?
A price is marked 25% above cost and then discounted by 20%. What is the final result on cost?
Which statement correctly uses a decimal interest rate?
A quantity rises by 20% and then falls by 20%. What percentage of the original remains?
A chart shows rates of 26% and 37% for two separate groups. What can be concluded without their sizes?
Practice Problems
- A 750 g mixture contains ingredients in the ratio 2:3:5. Find each ingredient’s mass, fraction and percentage. Then find the total mass of another mixture with the same proportions if its first ingredient weighs 240 g. Solution: There are 10 parts, so the shares are 1/5, 3/10 and 1/2, equal to 20%, 30% and 50%. Their masses in 750 g are 150 g, 225 g and 375 g. In the second mixture, 240 g is 20% of the whole, so total mass = 240 ÷ 0.20 = 1200 g.
- A seller buys an item for ₹1600, sets its marked price 25% above cost, then gives 10% off. A stated 5% tax applies to the discounted price. Find marked price, discounted price, final payment and profit percentage on cost before other expenses. Solution: Marked price = 1600 × 1.25 = ₹2000. Discounted price = 2000 × 0.9 = ₹1800. Tax = 1800 × 0.05 = ₹90, giving a ₹1890 payment. Profit = 1800 − 1600 = ₹200; profit on cost = 200/1600 × 100 = 12.5%.
- Compare total interest on ₹10,000 at 10% p.a. over three years with and without annual compounding. Explain the source of the difference. Solution: Simple interest = 10,000 × 0.10 × 3 = ₹3000. Compound amount = 10,000 × 1.1³ = ₹13,310, so compound interest = ₹3310. The extra ₹310 arises because interest from earlier years is included in the bases for later years. The simple model repeatedly uses the original ₹10,000.
- A group contains 200 people, 60% of whom are adults. Of the adults, 25% choose an activity. Of the remaining people, 50% choose it. Find the number and percentage of all people who choose it. Why is 25% + 50% not the answer? Solution: Adults = 0.6 × 200 = 120; remaining people = 80. Participating adults = 0.25 × 120 = 30; other participants = 0.5 × 80 = 40. Total participants = 70, giving 70/200 × 100 = 35%. The 25% and 50% refer to different subgroup bases and cannot be directly added.
- An item’s value falls by 10% per year for two years and then rises by 20%, becoming ₹9720. Find its original value, overall percentage change and the percentage rise needed from ₹9720 to restore the original. Solution: The combined factor is 0.9² × 1.2 = 0.972. Original value = 9720 ÷ 0.972 = ₹10,000. Overall loss in value = ₹280, which is 2.8% of the original. To restore ₹10,000 from ₹9720, the required increase is 280/9720 × 100 = 700/243%, approximately 2.88%. Its base is now ₹9720, so it differs from 2.8%.
Key Takeaways
• Identify the reference quantity that represents 100% before choosing an operation. • Fractions, decimals and percentages are interchangeable descriptions of a proportion. • Multiply for a direct share and divide to recover the original whole. • Distinguish a final percentage from a percentage increase or decrease. • Name the base in comparisons involving cost, selling price, discount, tax or revenue. • Repeated changes use updated bases and combine through multiplication. • Check estimates, units, rounding and whether a count is physically possible.
Previous · Lesson 11
Tricky Percentages
Next
End of chapter