Work Life & Job Hunting · SPI Prep Lab
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A high school has 800 pupils, of whom 40% are boys. How many boys are there?
Compared amount = base amount times proportion. The base amount is the 800 pupils in the school, so 800 × 0.4 = 320. Answering 480 gives the number of girls, from 100 − 40 = 60%. Answering 2000 comes from dividing, 800 ÷ 0.4, where you should multiply. Answering 3200 comes from misreading 0.4 as 4.
A course with a capacity of 250 received 200 applications. What percentage of the capacity is the number of applications?
Since it is 'of the capacity', the base amount is 250. 200 ÷ 250 = 0.8, that is 80%. Answering 125% comes from 250 ÷ 200, swapping the base amount and the compared amount. Answering 20% comes from working out 50 ÷ 250, the proportion of the 50 places left unfilled. Answering 50% reads the shortfall of 50 straight off as a percentage.
The price of an item rose from 1200 yen to 1500 yen. What is the rate of increase?
For a rate of change the base amount is the value before the change, 1200 yen. The rise is 300 yen, so 300 ÷ 1200 = 0.25, that is 25%. Answering 20% comes from 300 ÷ 1500, taking the price after the rise as the base amount. Answering 30% comes from reading only the '3' of the 300 yen rise and calling it three tenths, skipping the step of dividing by the base amount. Answering 125% comes from 1500 ÷ 1200, which gives the proportion of the whole rather than of the increase.
15000 yen is divided among three people A, B and C in the ratio 2:3:5. How much does B receive?
The parts of the ratio total 2 + 3 + 5 = 10. One part is 15000 ÷ 10 = 1500 yen, and B has three parts, so 1500 × 3 = 4500 yen. 3000 yen is A's share and 7500 yen is C's; 5000 yen is what you get by ignoring the ratio and splitting into three equal shares.
Given A:B = 3:4 and B:C = 6:5, which is A:C in its simplest whole-number form?
Line up the shared term B on the lowest common multiple of 4 and 6, which is 12. Then A:B = 9:12 and B:C = 12:10, so A:B:C = 9:12:10 and A:C = 9:10. Answering 3:5 just lines up the outer numbers without matching B. Answering 5:8 comes from cross-multiplying, 3 × 5 : 4 × 6 = 15:24.
You have read 120 pages of a book, which is 25% of the whole. How many pages does the book have in all?
Base amount = compared amount divided by proportion, so 120 ÷ 0.25 = 480 pages. Answering 30 comes from multiplying, 120 × 0.25. Answering 160 comes from 120 ÷ 0.75, taking 25% as the proportion left unread. Answering 600 comes from treating 25% as two tenths and calculating 120 ÷ 0.2.
Of the members of a club, 60% are second-years, and 25% of those second-years are women. What proportion of the whole membership are second-year women?
A proportion of a proportion is joined by multiplication: 0.6 × 0.25 = 0.15, that is 15%. Answering 85% adds 60 and 25; 35% subtracts 25 from 60; 1.5% shifts the decimal point of 0.15 by one place. Taking 100 members as a check gives 60 second-years, of whom 15 are women.
A shop had 2500 visitors last month, and this month the number fell by 12% from last month. How many visitors are there this month?
Value after = value before × (1 − 0.12) = 2500 × 0.88 = 2200. Answering 2800 works it as an increase, 2500 × 1.12. Answering 2488 subtracts 12 from 2500, confusing a proportion with an actual number. Answering 300 gives the fall itself, 2500 × 0.12, not this month's visitors.
Expressed as a percentage, what is 3割2分 (san-wari ni-bu), a rate written in the Japanese wari-bu notation?
One 割 (wari) is 10% and one 分 (bu) is 1%, so 3割2分 = 30% + 2% = 32%. Written as a decimal it is 0.32, and writing that straight off as a percentage gives 0.32%, while shifting the digits by one or two places gives 3.2% or 320%. Do not mix up the proportion itself with the percentage.
A drink is made by mixing concentrate and water in the ratio 2:7. When 150 mL of concentrate is used, how many millilitres of drink are made?
The concentrate is the 2 of the ratio, so one part is 150 ÷ 2 = 75 mL. The whole is 2 + 7 = 9 parts, so 75 × 9 = 675 mL. 525 mL is the water alone, 75 × 7; 1350 mL comes from treating the concentrate as one part and calculating 150 × 9; 300 mL comes from multiplying by the ratio figure directly, 150 × 2.
The elder and younger brother held money in the ratio 5:3. After the elder gave the younger 400 yen, the ratio became 3:2. How much did the elder brother hold at the start?
Write the starting amounts as 5x and 3x yen. After the transfer they are 5x − 400 and 3x + 400, so 2(5x − 400) = 3(3x + 400). From 10x − 800 = 9x + 1200 we get x = 2000, so the elder had 5 × 2000 = 10000 yen. 6000 yen is the younger brother's starting amount, 9600 yen is the elder's amount after the transfer, and 16000 yen is the pair's total.
A is 1.2 times B, and B is 1.5 times C. How many times C is A?
Take C as 1; then B is 1.5 and A is 1.5 × 1.2 = 1.8. Relations of 'times' are joined by multiplication. Answering 2.7 adds 1.2 and 1.5; 0.8 divides, 1.2 ÷ 1.5; 1.25 divides the other way, 1.5 ÷ 1.2.
A company has 640 staff, men and women together. 25% of the men and 40% of the women hold a certain qualification, and the holders number 205 in all. How many men are there?
Let the number of men be x; then 0.25x + 0.4(640 − x) = 205. Tidying the left side gives 256 − 0.15x = 205, so 0.15x = 51 and x = 340. As a check, 25% of 340 men is 85 and 40% of 300 women is 120, which totals 205. Answering 300 gives the number of women, and swapping the two percentages also lands on this value. Answering 85 gives the qualified men, and 256 comes from applying the women's rate to the whole, 640 × 0.4.
An item with a cost price of 800 yen is given a list price allowing a profit of 30% of the cost. What is the list price?
List price = cost price × (1 + profit rate) = 800 × 1.3 = 1040 yen. Answering 830 yen adds 30 straight onto 800, confusing a proportion with a sum of money. Answering 560 yen treats it as a discount, 800 × 0.7. Answering 240 yen gives the intended profit, 800 × 0.3, not the list price.
An item listed at 2500 yen is sold at 20% off. What is the selling price?
Selling price = list price × (1 − 0.2) = 2500 × 0.8 = 2000 yen. 500 yen is the discount, 2500 × 0.2, not the selling price. 2480 yen subtracts 20 from 2500, confusing a proportion with a sum of money. 3000 yen works it as a 20% increase, 2500 × 1.2.
An item with a cost price of 1200 yen was sold for 1500 yen. What percentage of the cost price is the profit?
The profit is 1500 − 1200 = 300 yen. Since it is 'of the cost price', the base amount is 1200 yen, so 300 ÷ 1200 = 0.25, that is 25%. Answering 20% comes from 300 ÷ 1500, taking the selling price as the base. Answering 125% gives 1500 ÷ 1200, the proportion of the whole selling price rather than of the profit. Answering 80% divides the other way, 1200 ÷ 1500.
An item costing 2000 yen was given a list price allowing a 25% profit, but as it did not sell it was sold at 20% off the list price. What is the profit per item?
The list price is 2000 × 1.25 = 2500 yen and the selling price is 2500 × 0.8 = 2000 yen, so the selling price equals the cost and the profit is 0 yen. The profit rate is based on the cost while the discount rate is based on the list price, and because the bases differ, a 25% rise followed by a 20% cut lands exactly back where it started. 500 yen is the profit if it sold at the list price. 100 yen applies 25 − 20 = 5% to the cost, 2000 × 0.05, and 125 yen applies the same 5% to the list price, 2500 × 0.05.
An item sold at 10% off the list price yielded a profit of 240 yen, which is 8% of the cost price. What is the cost price of this item?
8% of the cost is 240 yen, so the cost = 240 ÷ 0.08 = 3000 yen. Answering 2400 yen comes from dividing by the 10% discount, 240 ÷ 0.1. 3240 yen is the selling price, cost plus profit, 3000 + 240. 3600 yen is the list price found by dividing the selling price back by 0.9, 3240 ÷ 0.9; neither is the cost.
An item costing 4000 yen is to be given a list price such that even at 20% off the list price it still yields a profit of 20% of the cost. What should the list price be?
The selling price needed is 4000 × 1.2 = 4800 yen. The selling price is 0.8 of the list price, so the list price = 4800 ÷ 0.8 = 6000 yen. Checking, 6000 × 0.8 = 4800, a profit of 800 yen, which is 20% of the cost. 4800 yen is the selling price itself. 5000 yen ignores the profit and calculates 4000 ÷ 0.8. 5760 yen comes from multiplying by 1.2 to undo the 20% discount, 4800 × 1.2.
An item listed at 5000 yen was reduced by 20% and then a further 500 yen was taken off. What is the selling price?
Take 20% off first: 5000 × 0.8 = 4000 yen, then subtract 500 yen, giving 3500 yen. Answering 3600 yen reverses the order, (5000 − 500) × 0.8 = 3600; whether the percentage discount comes first or second changes the result. 4000 yen stops after the 20% cut and 4500 yen stops after the 500 yen cut.
An item listed at 8000 yen was reduced by 10%, and that price was reduced by a further 10%. What is the final selling price?
8000 × 0.9 = 7200 yen, then 7200 × 0.9 = 6480 yen. The base amount for the second cut is 7200 yen, so two successive 10% cuts do not add up to a 20% cut. 6400 yen lumps them into a single 20% cut, 8000 × 0.8. 7200 yen applies only one cut, and 1520 yen is the total reduction, 8000 − 6480.
An item costing 6000 yen was sold at 20% off the list price and made a loss of 600 yen. What is the list price of this item?
A loss of 600 yen means the selling price is 6000 − 600 = 5400 yen. The selling price is 0.8 of the list price, so the list price = 5400 ÷ 0.8 = 6750 yen. Checking, 6750 × 0.8 = 5400, which is 600 yen below the cost. 5400 yen is the selling price itself. 7500 yen leaves the loss out and calculates 6000 ÷ 0.8. 6600 yen adds the loss to the cost, 6000 + 600.
120 units were bought in at 300 yen each. 100 of them were sold at 450 yen each and the rest were cleared at 240 yen each. What is the total profit?
The total cost is 300 × 120 = 36000 yen. Takings are 450 × 100 + 240 × 20 = 45000 + 4800 = 49800 yen, so the profit is 49800 − 36000 = 13800 yen. 18000 yen is the profit if all 120 had sold at 450 yen, 54000 − 36000. 15000 yen counts only the profit on the 100 sold at 450 yen, 150 × 100. 49800 yen is the takings, not the profit.
An item was bought in at 900 yen each. The selling price is to be set so that 25% of the selling price is profit. What should the selling price be?
If 25% of the selling price is profit, the cost is 75% of the selling price. Selling price = 900 ÷ 0.75 = 1200 yen, and the profit of 300 yen is indeed 25% of it. 1125 yen works from the cost instead, 900 × 1.25. 675 yen uses the proportion the wrong way round, 900 × 0.75. 3600 yen divides the cost by the profit rate, 900 ÷ 0.25.
An item priced at 2400 yen before tax carries consumption tax of 10%. What is the tax-inclusive price?
Tax-inclusive price = pre-tax price × 1.1 = 2400 × 1.1 = 2640 yen. 2410 yen adds 10 to 2400, confusing a proportion with a sum of money. 2160 yen takes 10% off instead, 2400 × 0.9. 240 yen is the tax itself, not the tax-inclusive price.
An item costs 3300 yen including consumption tax of 10%. What is the pre-tax price?
The tax-inclusive price is 1.1 times the pre-tax price, so the pre-tax price = 3300 ÷ 1.1 = 3000 yen, and the tax is 300 yen. 2970 yen takes 10% off the tax-inclusive price, 3300 × 0.9. 3630 yen applies the tax again, 3300 × 1.1. 330 yen takes 10% of the tax-inclusive price as the tax, 3300 × 0.1.
An item was given a list price allowing a profit of 40% of the cost, and was then sold at 30% off the list price, making a loss of 240 yen per item. What is the cost price of this item?
Let the cost be x yen. The list price is 1.4x and the selling price is 1.4x × 0.7 = 0.98x. The loss is x − 0.98x = 0.02x, so 0.02x = 240 and x = 12000 yen. Checking, the list price is 16800 yen and the selling price 11760 yen, which is 240 yen below the cost. 2400 yen comes from treating it as a loss of 40% − 30% = 10% and calculating 240 ÷ 0.1. 16800 yen is the list price and 11760 yen the selling price.
40 g of salt was dissolved in 160 g of water. What is the concentration of this salt solution?
The whole solution is 40 + 160 = 200 g, so the concentration is 40 ÷ 200 = 0.2, that is 20%. The key point is that the denominator is the whole solution, not the water. Answering 25% puts the water in the denominator, 40 ÷ 160. 80% gives the proportion of water, 160 ÷ 200. 2% shifts the decimal point of 0.2 by one place.
How many grams of salt are contained in 450 g of an 8% salt solution?
Amount of salt = weight of solution × concentration = 450 × 0.08 = 36 g. 414 g is the water, 450 − 36, not the salt. 3.6 g shifts the decimal point by one place. 360 g comes from treating 8% as eight tenths, 450 × 0.8.
200 g of a 5% salt solution is mixed with 300 g of a 12% salt solution. What is the resulting concentration?
The salt is 200 × 0.05 = 10 g and 300 × 0.12 = 36 g, totalling 46 g, and the whole is 500 g, so 46 ÷ 500 = 0.092, that is 9.2%. Answering 8.5% takes the plain average of 5 and 12; since there is more of the 12% solution the result must lean that way, which shows the answer is wrong. 17% adds 5 and 12, and 7% subtracts 5 from 12.
100 g of water was evaporated from 500 g of an 8% salt solution. What is the concentration of the solution left?
The salt is 500 × 0.08 = 40 g and does not change when water evaporates. The whole is 500 − 100 = 400 g, so 40 ÷ 400 = 0.1, that is 10%. Answering 8% assumes the concentration stays the same when water is lost. 6.4% multiplies the ratio the wrong way round, 8 × 400 ÷ 500. 40% reads the 40 g of salt straight off as a percentage.
Water is to be added to 300 g of a 12% salt solution to bring it to 9%. How many grams of water must be added?
The salt is 300 × 0.12 = 36 g and does not change when water is added. At 9% the whole would be 36 ÷ 0.09 = 400 g, so the water added is 400 − 300 = 100 g. 400 g is the weight of the whole solution after adding, not the water added. 75 g applies the difference of the rates to the whole, 300 × (12 − 9) ÷ 12. 36 g is the salt.
A 6% salt solution and a 15% salt solution are to be mixed to make 450 g of a 10% salt solution. How many grams of the 6% solution are needed?
Let the 6% solution be x g; then 0.06x + 0.15(450 − x) = 45, so 0.09x = 22.5 and x = 250 g. The 15% solution is then 200 g, and the salt checks out as 15 + 30 = 45 g. Answering 200 g takes the balance-beam ratio the wrong way round: the ratio of weights is actually (15 − 10):(10 − 6) = 5:4, so there is more of the 6% side. 225 g is half of 450 g, and 45 g is the salt in the finished solution.
The price of an item rose by 20% in the first year and fell by 20% in the second year. After two years, what percentage of the original price is it?
Successive rises and falls are joined by multiplication: 1.2 × 0.8 = 0.96, that is 96%. The base amount for the second year's fall is the raised price, so the fall is the larger amount. Answering 100% assumes they cancel out. 144% treats both years as rises, 1.2 × 1.2. 40% subtracts, 1.2 − 0.8 = 0.4.
A company's sales rose by 20% last year over the year before, and fell this year by 25% from last year. When this year's sales are 54,000,000 yen, what were the sales two years ago?
This year = two years ago × 1.2 × 0.75 = two years ago × 0.9, so two years ago = 54,000,000 ÷ 0.9 = 60,000,000 yen. Checking, the chain runs 60,000,000 → 72,000,000 → 54,000,000. 48,600,000 yen multiplies where you should divide, 54,000,000 × 0.9. 64,800,000 yen undoes only the rise, 54,000,000 × 1.2. 72,000,000 yen is last year's sales.
A machine makes 210 parts in 6 minutes. At the same rate, how many minutes does it take to make 980?
It makes 210 ÷ 6 = 35 parts a minute, so 980 ÷ 35 = 28 minutes. Answering 35 minutes reads the per-minute figure of 35 straight off as minutes. Answering 22 minutes calculates only the 980 − 210 = 770 remaining parts, 770 ÷ 35, leaving out the time for the first 210.
An item priced at 180000 yen is bought with a deposit of 20% of the price and the rest paid in 12 equal monthly instalments. How much is each instalment?
The deposit is 180000 × 0.2 = 36000 yen, leaving 144000 yen. Each instalment is 144000 ÷ 12 = 12000 yen. 15000 yen forgets to subtract the deposit, 180000 ÷ 12. 3000 yen divides the 36000 yen deposit by 12 instead. 36000 yen is the deposit.
An item was bought with a deposit of 25000 yen and 20 monthly payments of 4500 yen. By how many yen does the total paid exceed the single-payment price of 105000 yen?
The monthly payments total 4500 × 20 = 90000 yen, and with the deposit the total is 115000 yen. 115000 − 105000 = 10000 yen more. 15000 yen forgets the deposit and compares 90000 yen with the single-payment price. 25000 yen is the deposit itself.
The same detergent is sold at shop A as 750 mL for 900 yen and at shop B as 1200 mL for 1380 yen. Which statement about the price per 100 mL is correct?
Shop A is 900 ÷ 7.5 = 120 yen and shop B is 1380 ÷ 12 = 115 yen, so per 100 mL shop B is 5 yen cheaper. 'Shop A is 5 yen cheaper' has the comparison the wrong way round. '50 yen cheaper' is the difference per 1000 mL, 1200 − 1150, mislabelled as per 100 mL. '480 yen cheaper' ignores the volumes and compares only the totals, 1380 − 900.
A shop makes 200 lunch boxes a day at a cost of 500 yen each. The list price is 800 yen, and any likely to be left over are sold before closing at 20% off the list price. On one day 150 sold at the list price and 50 at the discounted price. What was the profit that day?
The total cost is 500 × 200 = 100000 yen. The discounted price is 800 × 0.8 = 640 yen, so the takings are 800 × 150 + 640 × 50 = 120000 + 32000 = 152000 yen. The profit is 152000 − 100000 = 52000 yen. 60000 yen is the profit if all 200 had sold at the list price. 50000 yen comes from treating the 20% cut as 200 yen off and using a discounted price of 600 yen. 152000 yen is the takings.
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