Cross out first: size, units and multiplying back
Before you work out a percent question, cross out the options that cannot be right. Three quick checks do it: the size check (bigger or smaller than the number given?), the unit check (the same unit on top and bottom?) and the multiply-back check (does the option, times the multiplier, give the number you were given?). Often one option is left; if not, you solve with fewer choices and a check already built in.
Guess first
Guess first: after a 12% rise, a scooter costs ₹67,200. Which option can be its price before the rise? Cross out before you calculate.
Show answer
Correct answer: B
The price before a rise is smaller than ₹67,200, so ₹75,264 is out, and ₹7,200 is far too small to be a scooter's price before a 12% rise. Multiply back the two left: 60,000 × 1.12 = 67,200, a match; 59,136 × 1.12 = 66,232.32, not a match.
Option A: ₹75,264 raises ₹67,200 by another 12% (67,200 × 1.12 = 75,264). A price before a rise cannot be bigger than the price after it.
Option C: ₹59,136 takes 12% of ₹67,200 off (67,200 × 0.88 = 59,136). It fails the multiply-back check: 59,136 × 1.12 = 66,232.32.
Option D: ₹7,200 is the rise itself (67,200 − 60,000 = 7,200), not the old price.
Hint: Which options are the wrong size? Multiply back the rest by 1.12.
If you crossed out two options at once and then checked one product, you used the method of this lesson.
An everyday picture
Crossing out options works like a cashier checking a note before taking it.
Is it the right note at all? the size check
a price before a rise is smaller than the price now
Feel the security thread the multiply-back check
the option × the multiplier must give the number you were given
Rupees, not another currency the unit check
grams with grams, minutes with minutes, before you divide
A wrong option is usually made by a common slip, and each slip leaves a mark: the wrong size, the wrong unit, or a product that does not come back.
See it: the three checks on one question
The guess-first question again (after a 12% rise, the price is ₹67,200):
| Option | Size check: less than ₹67,200? | Multiply back: × 1.12 | Verdict |
|---|---|---|---|
| ₹75,264 | no, it is bigger | out | |
| ₹60,000 | yes | 60,000 × 1.12 = 67,200 | the answer |
| ₹59,136 | yes | 59,136 × 1.12 = 66,232.32 | out |
| ₹7,200 | yes, but it is only the rise | out |
The SSC notice (para 13.6) says the papers use the metric system of weights and measures. These are the conversions percent questions need most:
| Unit | Same as |
|---|---|
| 1 kg | 1,000 g |
| 1 m | 100 cm |
| 1 km | 1,000 m |
| 1 litre | 1,000 mL |
| 1 hour | 60 minutes |
| ₹1 | 100 paise |
flowchart TD A["Four options"] --> B["Size check: cross out the impossible ones"] B --> C["Unit check: the same unit on top and bottom"] C --> D["Multiply back the options that are left"] D --> E["The option that comes back exactly"]
Try it. 750 g are taken from a 2 kg bag of rice. What percent of the rice is that? Change both to grams first.
Find the rule
A tailor uses 90 cm from a 2.5 m length of cloth. What percent of the cloth did he use?
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Correct answer: C
Same unit first: 2.5 m = 250 cm. Then 90 ÷ 250 × 100 = 36%. Size check: 90 cm is less than 250 cm, so the answer must be under 100%.
Option A: 3,600% divides 90 by 2.5 without changing the metres to centimetres (90 ÷ 2.5 × 100 = 3,600). A part bigger than 100% fails the size check.
Option B: 3.6% changes 2.5 m into 2,500 cm (90 ÷ 2,500 × 100 = 3.6), but 1 m is 100 cm, not 1,000.
Option D: 277 7/9% divides the whole by the part (250 ÷ 90 × 100 = 277 7/9). The cloth used is the part, on top.
Hint: How many centimetres are in 2.5 m?
After a 30% fall, a share is worth ₹2,450. Which option can be its value before the fall?
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Correct answer: D
Before a fall the value was bigger than ₹2,450, so ₹1,715 and ₹1,050 are out. Multiply back the two left: 3,500 × 0.7 = 2,450, a match; 3,185 × 0.7 = 2,229.5, not a match.
Option A: ₹3,185 adds 30% of ₹2,450 (2,450 × 1.3 = 3,185). It fails the multiply-back check: 3,185 × 0.7 = 2,229.5.
Option B: ₹1,715 takes another 30% off (2,450 × 0.7 = 1,715). A value before a fall cannot be smaller than the value after it.
Option C: ₹1,050 is the fall itself (3,500 − 2,450 = 1,050), not the old value.
Hint: Before a fall, was the value bigger or smaller? Multiply back by 0.7.
Without working it out fully, which option can be 115% of 2,840?
Show answer
Correct answer: A
115% is a little more than the whole, so the answer is a little more than 2,840 and well under 2 × 2,840 = 5,680. Only 3,266 fits. Check: 2,840 × 1.15 = 3,266.
Option B: 326.6 is ten times too small (the decimal point slipped), and it is smaller than 2,840.
Option C: 6,106 adds 115% of 2,840 on top of 2,840 (2,840 + 3,266 = 6,106): more than twice the number.
Option D: 115 is the percent, not the amount.
Hint: Is 115% of a number a little more or a little less than the number?
Name the rule
- Size check: before a rise the value was smaller; before a fall it was bigger; a percent under 100% of a number is less than the number, and over 100% is more.
- Unit check: change both quantities to the same unit before dividing (1 kg = 1,000 g, 1 m = 100 cm, 1 litre = 1,000 mL, 1 hour = 60 minutes, ₹1 = 100 paise).
- Multiply-back check: an option is right only if the option × the multiplier gives the number in the question exactly.
- Order: size and units cross options out in seconds; multiply back only what is left.
Because exam options are rarely random. Each wrong option comes from a slip: going the wrong way, forgetting to convert, dividing the wrong way round. A slip makes a number of the wrong size or the wrong unit, and those are exactly what the checks catch.
Worked, half-done, solo
Question
300 g of a 1.2 kg cake is eaten. What percent of the cake is eaten?
Solution
- Same unit first
1.2 kg = 1,200 g.
- Size check
300 g is less than 1,200 g, so the answer is under 100%.
- Divide
300 ÷ 1,200 × 100 = 25%.
- Multiply back
25% of 1,200 g = 300 g.
Answer: 25%
Watch out: 300 ÷ 1.2 × 100 = 25,000% mixes grams with kilograms; the size check throws it out at once.
Now finish one that is half done.
600 mL are poured out of a 1.5 litre bottle. Same unit: 1.5 litres = 1,500 mL. What percent of the bottle is poured out?
Show answer
Correct answer: A
600 ÷ 1,500 × 100 = 40%. Check: 40% of 1,500 mL = 600 mL.
Option B: 250% divides the whole by the part (1,500 ÷ 600 × 100 = 250). More than 100% fails the size check.
Option C: 40,000% divides 600 by 1.5 without converting (600 ÷ 1.5 × 100 = 40,000).
Option D: 4% is ten times too small: the decimal point slipped one place.
Hint: Divide the part, 600 mL, by the whole, 1,500 mL.
And one on your own.
A ₹3,450 jacket gets a 20% discount, and then 5% GST is added to the sale price. Which option is the final price?
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Correct answer: D
A 20% discount and only 5% added back must leave the price below ₹3,450, so ₹3,622.50 is out. The multipliers: 3,450 × 0.8 × 1.05 = 2,898.
Option A: ₹2,932.50 adds the percents (−20 + 5 = −15) and takes 15% off ₹3,450 (3,450 × 0.85 = 2,932.50).
Option B: ₹2,760 stops after the discount (3,450 × 0.8 = 2,760) and forgets the GST.
Option C: ₹3,622.50 adds the GST to the marked price and forgets the discount (3,450 × 1.05 = 3,622.50).
Hint: Multiply by 0.8, then by 1.05.
The traps
- Skipped the size check: a value before a rise is smaller than the value now, and before a fall it is bigger.
- Mixed units: change both to the same unit before you divide.
- Wrong factor: 1 m is 100 cm, 1 kg is 1,000 g; a wrong factor moves the answer ten times or more.
- Trusted a near miss: an option that is close is not right until it multiplies back exactly.
- Crossed out too much: cross out only what is impossible, then check what is left.
Sana must find what percent 45 paise is of ₹1.80. She writes 45 ÷ 1.80 × 100 = 2,500%. What is right?
Show answer
Correct answer: A
45 paise is a small part of ₹1.80, so the answer must be under 100%: 2,500% fails the size check. In paise: ₹1.80 = 180 paise, and 45 ÷ 180 × 100 = 25%.
Option B: 2,500% divides paise by rupees; a part far bigger than the whole is impossible.
Option C: 400% puts the whole on top; the part, 45 paise, goes on top.
Option D: ₹1 is 100 paise, not 1,000, so 2.5% is ten times too small.
Hint: How many paise make ₹1.80?
Mixed check
In a 2-hour mock test, a student spends 54 minutes on the Quant part. What percent of the time is that?
Show answer
Correct answer: B
Same unit: 2 hours = 120 minutes. 54 ÷ 120 × 100 = 45%. Size check: under 100%, as it must be.
Option A: 2,700% divides 54 minutes by 2 hours without converting (54 ÷ 2 × 100 = 2,700).
Option C: 222 2/9% divides the whole by the part (120 ÷ 54 × 100 = 222 2/9).
Option D: 4.5% is ten times too small: the decimal point slipped one place.
Hint: How many minutes are in 2 hours?
From lesson 10:
About how much is 8.4% of 4,790?
Show answer
Correct answer: B
8.4% is close to 8⅓% = 1/12, and 4,790 is close to 4,800. 4,800 ÷ 12 = 400. The exact answer is 8.4% of 4,790 = 402.36.
Option A: 4,400 is the rest of 4,800 (4,800 − 400 = 4,400), not the part asked.
Option C: 40 is ten times too small: the decimal point slipped one place.
Option D: 576 divides 4,800 by 8⅓ (4,800 ÷ 8⅓ = 576).
Hint: 8⅓% is one twelfth.
Quick reference
Cross out first
- Size check
- before a rise: smaller than now; before a fall: bigger; under 100%: less than the whole after a 12% rise to ₹67,200, ₹75,264 is out
- Unit check
- the same unit on top and bottom 90 cm of 2.5 m: 90 ÷ 250 × 100 = 36%
- Metric factors
- 1 kg = 1,000 g; 1 m = 100 cm; 1 litre = 1,000 mL; 1 hour = 60 minutes; ₹1 = 100 paise 300 g of 1.2 kg: 300 ÷ 1,200 × 100 = 25%
- Multiply back
- option × multiplier = the number in the question 60,000 × 1.12 = 67,200
Why it works: wrong options come from common slips, and each slip leaves a mark: the wrong size, the wrong unit, or a product that does not come back.
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