UPSC CSE · CSAT · Numeracy and data interpretation · TOPIC LESSON
Track the reference whole and multiply change factors to solve proportional problems without switching bases.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original common-core lesson supports its declared paper scope. Optional subjects, qualifying language papers and the interview are outside this resource.
LEARN THROUGH ORIGINAL PRACTICE
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01 / UNDERSTAND & APPLY
For a nonnegative integer a and a positive integer b, the part-whole picture of a/b divides a whole into b equal parts and takes a such parts, allowing more than one whole when a exceeds b. More generally, a fraction denotes the quotient a÷b, with b nonzero; signed values are allowed. Multiplying numerator and denominator by the same nonzero number preserves that quotient. A fraction of one quantity cannot be added directly to a fraction of another without identifying what those quantities represent.
For numerical addition, convert to a common denominator:2/3+3/4=8/12+9/12=17/12. The denominator describes equal-sized units, so adding numerators before matching units is invalid. Multiplication instead combines proportions: two-thirds of three-quarters is(2×3)/(3×4)=1/2.
Division by a nonzero fraction multiplies by its reciprocal. Before using that rule in a word problem, identify whether the question asks a part of a quantity or how many parts fit into it. A fraction larger than 1 is legitimate; it represents more than one whole.
ORIGINAL PRACTICE · 1
The sum is 17/12; the product is 1/2.
The sum requires twelfths, giving 8/12+9/12. Multiplication gives 6/12 directly. Adding numerators and denominators to obtain 5/7 would perform neither operation correctly.
02 / UNDERSTAND & APPLY
A ratio a:b compares quantities in the same units or explicitly defined different units. Convert metres and centimetres before treating their numerical entries as comparable lengths. A part-to-part ratio 2:3 means the first share is 2/5 of their combined whole, not 2/3 of it.
To divide a total T in a positive ratio a:b:c, add the parts a+b+c, find one part T/(a+b+c), then multiply by each entry. Check that the shares sum to T and preserve the original ratios. Ratios alone do not determine actual amounts without a scale or total.
ORIGINAL WORKED EXAMPLE
Divide 840 units in the ratio 2:3:7. What percentage of the whole does the middle share receive?
Answer The shares are 140,210 and 490 units; the middle share is 25% of the total.
ORIGINAL PRACTICE · 1
3:1.
Convert 1.5 metres to 150 centimetres, then simplify 150:50. Comparing 1.5 directly with 50 would mix units.
03 / UNDERSTAND & APPLY
In direct proportion, y=kx for a fixed constant k: doubling x doubles y. In inverse proportion, xy=k: doubling x halves y. The relation must be supplied or justified by a model, not assumed because two quantities appear together.
For a fixed divisible job with equally productive workers and no interference, workers×time can remain constant. If more workers create crowding or the task cannot be divided, the model may fail. Similarly, cost may not scale directly with quantity when fixed fees or bulk discounts are present.
State what is held fixed, such as total work, working hours per day and individual productivity. The formula is a compact representation of those conditions, not a guarantee about every real workplace.
ORIGINAL PRACTICE · 1
9 days.
Total work is 12×15=180 worker-days. At 20 workers, time=180/20=9. This inverse relationship depends on divisible work and unchanged productivity.
04 / UNDERSTAND & APPLY
p percent means p/100 of a specified base. Relative change is(new−old)/old×100%, so the old value is the denominator. If the old value is zero, this formula is undefined; report the absolute change or another clearly defined measure instead of a conventional percentage increase from zero.
A change between percentages can be described in percentage points or relative percent. Moving from 30% to 36% is 6 percentage points, but relative to 30% it is 6/30=20%. The two descriptions answer different questions.
Identify the base in language before calculating. “Twenty percent of visitors” is not the same denominator as “twenty percent of complaints”. A percentage can exceed 100 when comparing a quantity larger than the chosen base; this is not automatically an error.
ORIGINAL PRACTICE · 1
It rises 6 percentage points, which is a 20% relative increase.
Subtract 36−30 for points. Divide the 6-point increase by the original 30-point share to obtain 0.2=20%. Using 36 as the denominator would describe a different comparison.
05 / UNDERSTAND & APPLY
An increase of r% multiplies by 1+r/100; a decrease multiplies by 1−r/100. Successive changes act on successive bases, so multiply factors. Equal percentage increases and decreases do not cancel because the second percentage applies to the changed amount.
To reverse a percentage change, divide the final amount by its factor. Subtracting the same percentage from the final amount uses the wrong base. Check the recovered original by applying the forward factor.
Under an ideal multiplicative model with the same factors and no intermediate rounding, their order does not change the final product. In real applications, rounding, caps or quantity-dependent rules can make order matter; use the stipulated convention.
ORIGINAL WORKED EXAMPLE
A price rises 20% and then falls 20%. Find the overall percentage change.
Answer A 4% net decrease.
ORIGINAL WORKED EXAMPLE
After a 15% decrease, a quantity is 680. What was its original value?
Answer 800. Taking 15% off 680 again would reverse neither the operation nor its base.
ORIGINAL PRACTICE · 1
32.5%.
The retained fraction is 0.75×0.90=0.675. Thus 67.5% of the marked price remains, so the discount is 100−67.5=32.5%, not 35%.
06 / UNDERSTAND & APPLY
In the conventional arithmetic used here, profit is selling price minus cost, and profit percentage uses cost as its base. Loss percentage also uses cost. Discount is marked price minus selling price, and discount percentage uses marked price. State another convention explicitly if a problem supplies it.
A single transaction can therefore have a 20% discount and a 20% profit without contradiction, because the bases differ. Recovering cost from a profit percentage divides selling price by the profit factor; subtracting the same percentage from selling price is a common wrong-base shortcut. These are hypothetical arithmetic models, not advice about actual purchases or investments.
ORIGINAL WORKED EXAMPLE
An item has marked price 1500, receives a 20% discount and has cost 1000. Find the selling price, profit and profit percentage on cost.
Answer Selling price 1200, profit 200 and profit percentage 20% on cost.
ORIGINAL PRACTICE · 1
600.
Selling price=1.25×cost, so cost=750/1.25=600. Check the 150 profit is 25% of 600, not 25% of 750.
07 / UNDERSTAND & APPLY
Simple interest on principal P at annual rate r for t years is P×r×t when r is written as a fraction. It applies each year’s interest to the original principal. With annual compounding for an integer n years, amount=P(1+r)^n because each completed year’s interest joins the next year’s base.
Always align the rate period, duration and compounding convention. An annual rate does not by itself specify how a part-year or monthly calculation should be handled. This lesson uses fully stated numerical models and makes no claim about an actual product’s fees, taxes or terms.
Compound growth is also a general mathematical idea outside money. A hypothetical population or quantity can change by a stated factor per period. Avoid presenting that illustrative number as a current factual statistic.
ORIGINAL WORKED EXAMPLE
A hypothetical principal of 5000 earns 10% per year for two complete years. Compare simple interest with annual compounding, with no fees, tax or rounding.
Answer Simple amount 6000 and compound amount 6050; the compound model yields 50 more over the stated two-year period.
ORIGINAL PRACTICE · 1
840.
Multiply 800 by 1.05, or add 0.05×800=40. This is a supplied hypothetical growth model, not a claim about a real population.
08 / UNDERSTAND & APPLY
For fractions with positive denominators, cross multiplication preserves order: a/b<c/d exactly when ad<bc. If denominators may be negative, first normalize signs or track sign reversals. A zero denominator makes the fraction undefined.
An estimate can identify an implausible answer: a 25% discount should leave three-quarters, not one-quarter, of a positive price. But a close comparison may need exact arithmetic. Use a common denominator or factors rather than a prematurely rounded decimal.
At the end, name the reference whole again. If a problem asks the percentage of the final amount rather than the original, a correct calculation using the old base is still the wrong answer.
ORIGINAL PRACTICE · 1
5/8.
Use common denominator 24:7/12=14/24 and 5/8=15/24. Positive denominators also permit cross products 7×8=56 and 5×12=60, giving the same order.
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