4.2 Percents, money, and rates

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4. Arithmetic Reasoning

Percents, money, and rates

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Percent questions are among the most common Arithmetic Reasoning problems. Nearly all of them come down to one relationship among three numbers: the part, the whole, and the percent.

The percent equation

$part = percent \times whole$

Write the percent as a decimal before you multiply: $35%$ is $0.35$ , $8%$ is $0.08$ , and $150%$ is $1.5$ .

Each question gives you two of the three numbers and asks for the third.

Finding the part: multiply. $15%$ of $60$ is $0.15 \times 60 = 9$ .
Finding the whole: divide the part by the percent. If $12$ is $30%$ of a number, the number is $12 \div 0.30 = 40$ .
Finding the percent: divide the part by the whole. $18$ out of $24$ is $18 \div 24 = 0.75$ , or $75%$ .

When you’re finding a percent, the whole (the total, or the number after “of”) goes on the bottom. Dividing the other way, $24 \div 18$ , is the most common mistake, and its result is usually one of the answer choices.

Percent increase and decrease

To find a percent change, divide the amount of change by the original amount:

$percent change = \frac{new - original}{original}$

A negative result is a decrease. A price that falls from $$100$ to $$80$ changes by $\frac{- 20}{100} = - 20%$ , so it decreased by $20%$ .

Example:

A worker’s hourly pay rises from $$16$ to $$20$ . By what percent did it increase?

A. $4%$
B. $20%$
C. $25%$
D. $80%$

The pay went up by $$20 - $16 = $4$ . Divide by the original pay:

$\frac{4}{16} = 0.25 = 25%$

Choice B divides by the new pay ( $4 \div 20$ ), which is the trap. A percent change is always measured from where the amount started.

Answer: C

Discounts, sales tax, tips, and commission

A discount comes off the price. Sales tax and a tip are added to it. Each is a percent of the price it’s figured on.

A quick way to handle either one is to multiply by what’s left. $25%$ off means you pay $75%$ of the price, so multiply by $0.75$ . An $8%$ tax means you pay $108%$ , so multiply by $1.08$ .

Example:

A pair of boots costs $$120$ . They’re on sale for $25%$ off, and sales tax is $5%$ . What is the total cost?

A. $$90.00$
B. $$94.50$
C. $$96.00$
D. $$126.00$

First, take off the discount: $$120 \times 0.75 = $90$ .

The tax is figured on the price you actually pay, which is $$90$ : $$90 \times 0.05 = $4.50$ .

Add it on: $$90 + $4.50 = $94.50$ .

Answer: B

Two discounts in a row don’t add. The second discount is taken off the already-reduced price. An item marked $20%$ off and then another $10%$ off isn’t $30%$ off:

$$200 \times 0.80 = $160, $160 \times 0.90 = $144$

The final price is $$144$ , not the $$140$ you’d get by taking $30%$ off.

To compare a percent discount with a dollar discount, turn the percent into dollars first. On a $$60$ item, $15%$ off is $0.15 \times $60 = $9$ off, which beats a $$8$ coupon by $$1$ .

To split a bill with a tip, add the tip first, then divide. A $$72$ bill with a $20%$ tip comes to $$72 \times 1.20 = $86.40$ , so $4$ friends each pay $$86.40 \div 4 = $21.60$ .

A commission is a percent of what a salesperson sells. At a $6%$ commission, $$4, 500$ in sales earns $0.06 \times $4, 500 = $270$ .

Simple interest

Simple interest is paid only on the original amount, called the principal:

$I = P \times r \times t$

Here $P$ is the principal, $r$ is the yearly interest rate written as a decimal, and $t$ is the time in years.

Example:

How much simple interest is earned on $$2, 000$ invested at $4%$ for $3$ years?

A. $$80$
B. $$240$
C. $$2, 240$
D. $$2, 400$

$I = 2, 000 \times 0.04 \times 3 = 240$

The other choices are common slips. $$80$ is one year of interest. $$2, 240$ is the account’s balance (principal plus interest), which answers a different question. $$2, 400$ comes from writing $4%$ as $0.4$ .

Answer: B

Depreciation

When something loses a percent of its value every year, each year’s loss is figured on what it was worth at the start of that year, not on the original price.

Example:

A $$1, 000$ machine loses $20%$ of its value each year. What is it worth after $2$ years?

Each year it keeps $80%$ of its value:

$$1, 000 \times 0.80 = $800 after one year$

$$800 \times 0.80 = $640 after two years$

Subtracting $$200$ twice gives $$600$ , which is wrong: in the second year the machine loses $20%$ of $$800$ , only $$160$ .

Answer: $$640$

Knowledge check

A jacket normally costs $$80$ . It’s on sale for $25%$ off. What is the sale price?

A. $$20$
B. $$55$
C. $$60$
D. $$65$

(spoiler)

$25%$ of $$80$ is $0.25 \times $80 = $20$ off the original price. Subtract the discount: $$80 - $20 = $60$ .

Answer: C. $$60$

Rates, distance, and work

A rate compares two different kinds of quantities: miles per hour, pages per minute, dollars per pound. “Per” means “for each one.”

Unit rates and unit prices

To find a unit rate, divide the total amount by the number of units.

Example:

A car travels $270$ miles on $9$ gallons of gas. How many miles per gallon does it get?

$270 \div 9 = 30$

Answer: $30$ miles per gallon

Once you know the rate, multiply to scale up. At $30$ miles per gallon, $12$ gallons take the car $30 \times 12 = 360$ miles.

A unit price is the cost of one unit, and it tells you which size is the better buy.

Example:

An $18$ -ounce box of cereal costs $$4.50$ , and a $24$ -ounce box costs $$5.52$ . Which is the better buy?

$$4.50 \div 18 = $0.25 per ounce$

$$5.52 \div 24 = $0.23 per ounce$

Answer: The $24$ -ounce box, which costs $2$ cents less per ounce.

Distance, rate, and time

The distance formula

$distance = rate \times time$

Rearranged: $time = distance \div rate$ , and $rate = distance \div time$ .

The units have to match. A rate in miles per hour needs a time in hours, so convert minutes first: divide by $60$ . Thirty minutes is $\frac{30}{60} = 0.5$ hour, and $45$ minutes is $0.75$ hour.

Example:

A cyclist rides for $2$ hours and $30$ minutes at a steady $14$ miles per hour. How far does the cyclist travel?

A. $28$ miles
B. $32.2$ miles
C. $35$ miles
D. $42$ miles

$2$ hours and $30$ minutes is $2.5$ hours, not $2.3$ hours.

$14 \times 2.5 = 35$

Answer: C

Averaging two speeds is a trap. When a trip has two legs at different speeds, find the time for each leg separately.

Example:

A driver travels $120$ miles at $60$ miles per hour, then another $120$ miles at $40$ miles per hour. What is the total travel time?

A. $4$ hours
B. $4.8$ hours
C. $5$ hours
D. $6$ hours

$120 \div 60 = 2 hours, 120 \div 40 = 3 hours$

The trip takes $2 + 3 = 5$ hours. Choice B averages the two speeds to $50$ miles per hour and divides $240$ miles by it. That’s wrong because the driver spends more time at the slower speed: the true average speed is $240 \div 5 = 48$ miles per hour.

Answer: C

When two travelers move apart in opposite directions, the distance between them grows by the sum of their speeds. Two cars leaving the same spot at $50$ and $60$ miles per hour are $(50 + 60) \times 3 = 330$ miles apart after $3$ hours. When one follows the other in the same direction, the gap changes by the difference in their speeds.

Work problems

When two people or machines work at the same time, their rates add.

Example:

Printer A prints $20$ pages per minute, and Printer B prints $30$ pages per minute. Working at the same time, how long do they take to print $400$ pages?

Together they print $20 + 30 = 50$ pages per minute:

$400 \div 50 = 8$

Answer: $8$ minutes

If a problem gives the time each worker needs for the whole job, turn each time into a rate first. A painter who finishes a room in $6$ hours paints $\frac{1}{6}$ of the room per hour; one who takes $3$ hours paints $\frac{1}{3}$ . Together they paint $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ of the room per hour, so the room takes $2$ hours.

When a problem changes both the number of machines and the hours, find what one machine does in one hour.

Example:

A factory makes $240$ parts in $4$ hours with $3$ machines running. At the same rate, how many parts can $5$ machines make in $6$ hours?

$240 \div 4 = 60$ parts per hour, from $3$ machines. That’s $60 \div 3 = 20$ parts per machine per hour.

$20 \times 5 \times 6 = 600$

Answer: $600$ parts

The percent equation

$part = percent \times whole$ ; convert percent to decimal first ( $35% = 0.35$ )
Finding part: multiply percent × whole
Finding whole: divide part by percent ( $12 \div 0.30 = 40$ )
Finding percent: divide part by whole, with whole (the “of” number) on bottom — dividing backwards is the common trap

Percent increase/decrease

$percent change = \frac{new - original}{original}$
Negative result = decrease
Always divide by the original value, not the new one (common trap)

Discounts, tax, tips, commission

Discount: pay (100% − discount%) of price; Tax/tip: pay (100% + rate%) of price
Multiply-by-remaining-percent shortcut: 25% off → ×0.75; 8% tax → ×1.08
Tax/tip applied to price after discount, not original price
Multiple discounts don’t add — apply sequentially (20% then 10% off ≠ 30% off)
To compare % discount vs. $ discount, convert % to dollars first
Splitting a tip: add tip to total first, then divide among people
Commission = percent × sales amount

Simple interest

Formula: $I = P \times r \times t$ (P=principal, r=decimal rate/year, t=years)
Watch for traps: one year’s interest only, confusing interest with final balance, or writing % wrong (4% ≠ 0.4)

Depreciation

Each year’s loss is calculated on the current value, not the original price
Multiply by remaining percent each year sequentially (e.g., ×0.80 repeatedly for 20% annual loss)
Don’t just subtract the same dollar amount each year

Rates and unit prices

Unit rate = total ÷ number of units (e.g., miles per gallon)
Unit price = cost ÷ units; lower unit price = better buy
Scale up by multiplying known rate by new quantity

Distance, rate, time

$distance = rate \times time$ ; rearrange for rate or time as needed
Units must match — convert minutes to hours (divide by 60) before using hourly rates
Multi-leg trips: compute time for each leg separately, don’t average speeds directly
True average speed = total distance ÷ total time (not simple average of two speeds)
Opposite directions: combined speed = sum of rates; same direction (chasing): relative speed = difference of rates

Work problems

Rates add when working simultaneously: combined rate = rate₁ + rate₂
Convert “time to complete alone” into a rate (fraction of job per hour) before adding
For machines/hours combo problems: find output per single unit per hour, then scale by new machines × new hours

Next | 4.3 Ratios, measurement, and probability

Percents, money, and rates

Percent questions are among the most common Arithmetic Reasoning problems. Nearly all of them come down to one relationship among three numbers: the part, the whole, and the percent.

The percent equation

$part = percent \times whole$

Write the percent as a decimal before you multiply: $35%$ is $0.35$ , $8%$ is $0.08$ , and $150%$ is $1.5$ .

Each question gives you two of the three numbers and asks for the third.

Finding the part: multiply. $15%$ of $60$ is $0.15 \times 60 = 9$ .
Finding the whole: divide the part by the percent. If $12$ is $30%$ of a number, the number is $12 \div 0.30 = 40$ .
Finding the percent: divide the part by the whole. $18$ out of $24$ is $18 \div 24 = 0.75$ , or $75%$ .

Percent increase and decrease

To find a percent change, divide the amount of change by the original amount:

$percent change = \frac{new - original}{original}$

A negative result is a decrease. A price that falls from $$100$ to $$80$ changes by $\frac{- 20}{100} = - 20%$ , so it decreased by $20%$ .

Example:

A worker’s hourly pay rises from $$16$ to $$20$ . By what percent did it increase?

A. $4%$
B. $20%$
C. $25%$
D. $80%$

The pay went up by $$20 - $16 = $4$ . Divide by the original pay:

$\frac{4}{16} = 0.25 = 25%$

Choice B divides by the new pay ( $4 \div 20$ ), which is the trap. A percent change is always measured from where the amount started.

Answer: C

Discounts, sales tax, tips, and commission

A discount comes off the price. Sales tax and a tip are added to it. Each is a percent of the price it’s figured on.

A quick way to handle either one is to multiply by what’s left. $25%$ off means you pay $75%$ of the price, so multiply by $0.75$ . An $8%$ tax means you pay $108%$ , so multiply by $1.08$ .

Example:

A pair of boots costs $$120$ . They’re on sale for $25%$ off, and sales tax is $5%$ . What is the total cost?

A. $$90.00$
B. $$94.50$
C. $$96.00$
D. $$126.00$

First, take off the discount: $$120 \times 0.75 = $90$ .

The tax is figured on the price you actually pay, which is $$90$ : $$90 \times 0.05 = $4.50$ .

Add it on: $$90 + $4.50 = $94.50$ .

Answer: B

Two discounts in a row don’t add. The second discount is taken off the already-reduced price. An item marked $20%$ off and then another $10%$ off isn’t $30%$ off:

$$200 \times 0.80 = $160, $160 \times 0.90 = $144$

The final price is $$144$ , not the $$140$ you’d get by taking $30%$ off.

To compare a percent discount with a dollar discount, turn the percent into dollars first. On a $$60$ item, $15%$ off is $0.15 \times $60 = $9$ off, which beats a $$8$ coupon by $$1$ .

To split a bill with a tip, add the tip first, then divide. A $$72$ bill with a $20%$ tip comes to $$72 \times 1.20 = $86.40$ , so $4$ friends each pay $$86.40 \div 4 = $21.60$ .

A commission is a percent of what a salesperson sells. At a $6%$ commission, $$4, 500$ in sales earns $0.06 \times $4, 500 = $270$ .

Simple interest

Simple interest is paid only on the original amount, called the principal:

$I = P \times r \times t$

Here $P$ is the principal, $r$ is the yearly interest rate written as a decimal, and $t$ is the time in years.

Example:

How much simple interest is earned on $$2, 000$ invested at $4%$ for $3$ years?

A. $$80$
B. $$240$
C. $$2, 240$
D. $$2, 400$

$I = 2, 000 \times 0.04 \times 3 = 240$

The other choices are common slips. $$80$ is one year of interest. $$2, 240$ is the account’s balance (principal plus interest), which answers a different question. $$2, 400$ comes from writing $4%$ as $0.4$ .

Answer: B

Depreciation

When something loses a percent of its value every year, each year’s loss is figured on what it was worth at the start of that year, not on the original price.

Example:

A $$1, 000$ machine loses $20%$ of its value each year. What is it worth after $2$ years?

Each year it keeps $80%$ of its value:

$$1, 000 \times 0.80 = $800 after one year$

$$800 \times 0.80 = $640 after two years$

Subtracting $$200$ twice gives $$600$ , which is wrong: in the second year the machine loses $20%$ of $$800$ , only $$160$ .

Answer: $$640$

Knowledge check

A jacket normally costs $$80$ . It’s on sale for $25%$ off. What is the sale price?

A. $$20$
B. $$55$
C. $$60$
D. $$65$

(spoiler)

$25%$ of $$80$ is $0.25 \times $80 = $20$ off the original price. Subtract the discount: $$80 - $20 = $60$ .

Answer: C. $$60$

Rates, distance, and work

A rate compares two different kinds of quantities: miles per hour, pages per minute, dollars per pound. “Per” means “for each one.”

Unit rates and unit prices

To find a unit rate, divide the total amount by the number of units.

Example:

A car travels $270$ miles on $9$ gallons of gas. How many miles per gallon does it get?

$270 \div 9 = 30$

Answer: $30$ miles per gallon

Once you know the rate, multiply to scale up. At $30$ miles per gallon, $12$ gallons take the car $30 \times 12 = 360$ miles.

A unit price is the cost of one unit, and it tells you which size is the better buy.

Example:

An $18$ -ounce box of cereal costs $$4.50$ , and a $24$ -ounce box costs $$5.52$ . Which is the better buy?

$$4.50 \div 18 = $0.25 per ounce$

$$5.52 \div 24 = $0.23 per ounce$

Answer: The $24$ -ounce box, which costs $2$ cents less per ounce.

Distance, rate, and time

The distance formula

$distance = rate \times time$

Rearranged: $time = distance \div rate$ , and $rate = distance \div time$ .

Example:

A cyclist rides for $2$ hours and $30$ minutes at a steady $14$ miles per hour. How far does the cyclist travel?

A. $28$ miles
B. $32.2$ miles
C. $35$ miles
D. $42$ miles

$2$ hours and $30$ minutes is $2.5$ hours, not $2.3$ hours.

$14 \times 2.5 = 35$

Answer: C

Averaging two speeds is a trap. When a trip has two legs at different speeds, find the time for each leg separately.

Example:

A driver travels $120$ miles at $60$ miles per hour, then another $120$ miles at $40$ miles per hour. What is the total travel time?

A. $4$ hours
B. $4.8$ hours
C. $5$ hours
D. $6$ hours

$120 \div 60 = 2 hours, 120 \div 40 = 3 hours$

The trip takes $2 + 3 = 5$ hours. Choice B averages the two speeds to $50$ miles per hour and divides $240$ miles by it. That’s wrong because the driver spends more time at the slower speed: the true average speed is $240 \div 5 = 48$ miles per hour.

Answer: C

Work problems

When two people or machines work at the same time, their rates add.

Example:

Printer A prints $20$ pages per minute, and Printer B prints $30$ pages per minute. Working at the same time, how long do they take to print $400$ pages?

Together they print $20 + 30 = 50$ pages per minute:

$400 \div 50 = 8$

Answer: $8$ minutes

When a problem changes both the number of machines and the hours, find what one machine does in one hour.

Example:

A factory makes $240$ parts in $4$ hours with $3$ machines running. At the same rate, how many parts can $5$ machines make in $6$ hours?

$240 \div 4 = 60$ parts per hour, from $3$ machines. That’s $60 \div 3 = 20$ parts per machine per hour.

$20 \times 5 \times 6 = 600$

Answer: $600$ parts

Achievable ASVAB

4. Arithmetic Reasoning

Percents, money, and rates

8 min read

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Discuss

Feedback

Percent questions are among the most common Arithmetic Reasoning problems. Nearly all of them come down to one relationship among three numbers: the part, the whole, and the percent.

The percent equation

$part = percent \times whole$

Write the percent as a decimal before you multiply: $35%$ is $0.35$ , $8%$ is $0.08$ , and $150%$ is $1.5$ .

Each question gives you two of the three numbers and asks for the third.

Finding the part: multiply. $15%$ of $60$ is $0.15 \times 60 = 9$ .
Finding the whole: divide the part by the percent. If $12$ is $30%$ of a number, the number is $12 \div 0.30 = 40$ .
Finding the percent: divide the part by the whole. $18$ out of $24$ is $18 \div 24 = 0.75$ , or $75%$ .

Percent increase and decrease

To find a percent change, divide the amount of change by the original amount:

$percent change = \frac{new - original}{original}$

A negative result is a decrease. A price that falls from $$100$ to $$80$ changes by $\frac{- 20}{100} = - 20%$ , so it decreased by $20%$ .

Example:

A worker’s hourly pay rises from $$16$ to $$20$ . By what percent did it increase?

A. $4%$
B. $20%$
C. $25%$
D. $80%$

The pay went up by $$20 - $16 = $4$ . Divide by the original pay:

$\frac{4}{16} = 0.25 = 25%$

Choice B divides by the new pay ( $4 \div 20$ ), which is the trap. A percent change is always measured from where the amount started.

Answer: C

Discounts, sales tax, tips, and commission

A discount comes off the price. Sales tax and a tip are added to it. Each is a percent of the price it’s figured on.

A quick way to handle either one is to multiply by what’s left. $25%$ off means you pay $75%$ of the price, so multiply by $0.75$ . An $8%$ tax means you pay $108%$ , so multiply by $1.08$ .

Example:

A pair of boots costs $$120$ . They’re on sale for $25%$ off, and sales tax is $5%$ . What is the total cost?

A. $$90.00$
B. $$94.50$
C. $$96.00$
D. $$126.00$

First, take off the discount: $$120 \times 0.75 = $90$ .

The tax is figured on the price you actually pay, which is $$90$ : $$90 \times 0.05 = $4.50$ .

Add it on: $$90 + $4.50 = $94.50$ .

Answer: B

Two discounts in a row don’t add. The second discount is taken off the already-reduced price. An item marked $20%$ off and then another $10%$ off isn’t $30%$ off:

$$200 \times 0.80 = $160, $160 \times 0.90 = $144$

The final price is $$144$ , not the $$140$ you’d get by taking $30%$ off.

To compare a percent discount with a dollar discount, turn the percent into dollars first. On a $$60$ item, $15%$ off is $0.15 \times $60 = $9$ off, which beats a $$8$ coupon by $$1$ .

To split a bill with a tip, add the tip first, then divide. A $$72$ bill with a $20%$ tip comes to $$72 \times 1.20 = $86.40$ , so $4$ friends each pay $$86.40 \div 4 = $21.60$ .

A commission is a percent of what a salesperson sells. At a $6%$ commission, $$4, 500$ in sales earns $0.06 \times $4, 500 = $270$ .

Simple interest

Simple interest is paid only on the original amount, called the principal:

$I = P \times r \times t$

Here $P$ is the principal, $r$ is the yearly interest rate written as a decimal, and $t$ is the time in years.

Example:

How much simple interest is earned on $$2, 000$ invested at $4%$ for $3$ years?

A. $$80$
B. $$240$
C. $$2, 240$
D. $$2, 400$

$I = 2, 000 \times 0.04 \times 3 = 240$

The other choices are common slips. $$80$ is one year of interest. $$2, 240$ is the account’s balance (principal plus interest), which answers a different question. $$2, 400$ comes from writing $4%$ as $0.4$ .

Answer: B

Depreciation

When something loses a percent of its value every year, each year’s loss is figured on what it was worth at the start of that year, not on the original price.

Example:

A $$1, 000$ machine loses $20%$ of its value each year. What is it worth after $2$ years?

Each year it keeps $80%$ of its value:

$$1, 000 \times 0.80 = $800 after one year$

$$800 \times 0.80 = $640 after two years$

Subtracting $$200$ twice gives $$600$ , which is wrong: in the second year the machine loses $20%$ of $$800$ , only $$160$ .

Answer: $$640$

Knowledge check

A jacket normally costs $$80$ . It’s on sale for $25%$ off. What is the sale price?

A. $$20$
B. $$55$
C. $$60$
D. $$65$

(spoiler)

$25%$ of $$80$ is $0.25 \times $80 = $20$ off the original price. Subtract the discount: $$80 - $20 = $60$ .

Answer: C. $$60$

Rates, distance, and work

A rate compares two different kinds of quantities: miles per hour, pages per minute, dollars per pound. “Per” means “for each one.”

Unit rates and unit prices

To find a unit rate, divide the total amount by the number of units.

Example:

A car travels $270$ miles on $9$ gallons of gas. How many miles per gallon does it get?

$270 \div 9 = 30$

Answer: $30$ miles per gallon

Once you know the rate, multiply to scale up. At $30$ miles per gallon, $12$ gallons take the car $30 \times 12 = 360$ miles.

A unit price is the cost of one unit, and it tells you which size is the better buy.

Example:

An $18$ -ounce box of cereal costs $$4.50$ , and a $24$ -ounce box costs $$5.52$ . Which is the better buy?

$$4.50 \div 18 = $0.25 per ounce$

$$5.52 \div 24 = $0.23 per ounce$

Answer: The $24$ -ounce box, which costs $2$ cents less per ounce.

Distance, rate, and time

The distance formula

$distance = rate \times time$

Rearranged: $time = distance \div rate$ , and $rate = distance \div time$ .

Example:

A cyclist rides for $2$ hours and $30$ minutes at a steady $14$ miles per hour. How far does the cyclist travel?

A. $28$ miles
B. $32.2$ miles
C. $35$ miles
D. $42$ miles

$2$ hours and $30$ minutes is $2.5$ hours, not $2.3$ hours.

$14 \times 2.5 = 35$

Answer: C

Averaging two speeds is a trap. When a trip has two legs at different speeds, find the time for each leg separately.

Example:

A driver travels $120$ miles at $60$ miles per hour, then another $120$ miles at $40$ miles per hour. What is the total travel time?

A. $4$ hours
B. $4.8$ hours
C. $5$ hours
D. $6$ hours

$120 \div 60 = 2 hours, 120 \div 40 = 3 hours$

The trip takes $2 + 3 = 5$ hours. Choice B averages the two speeds to $50$ miles per hour and divides $240$ miles by it. That’s wrong because the driver spends more time at the slower speed: the true average speed is $240 \div 5 = 48$ miles per hour.

Answer: C

Work problems

When two people or machines work at the same time, their rates add.

Example:

Printer A prints $20$ pages per minute, and Printer B prints $30$ pages per minute. Working at the same time, how long do they take to print $400$ pages?

Together they print $20 + 30 = 50$ pages per minute:

$400 \div 50 = 8$

Answer: $8$ minutes

When a problem changes both the number of machines and the hours, find what one machine does in one hour.

Example:

A factory makes $240$ parts in $4$ hours with $3$ machines running. At the same rate, how many parts can $5$ machines make in $6$ hours?

$240 \div 4 = 60$ parts per hour, from $3$ machines. That’s $60 \div 3 = 20$ parts per machine per hour.

$20 \times 5 \times 6 = 600$

Answer: $600$ parts

The percent equation

$part = percent \times whole$ ; convert percent to decimal first ( $35% = 0.35$ )
Finding part: multiply percent × whole
Finding whole: divide part by percent ( $12 \div 0.30 = 40$ )
Finding percent: divide part by whole, with whole (the “of” number) on bottom — dividing backwards is the common trap

Percent increase/decrease

$percent change = \frac{new - original}{original}$
Negative result = decrease
Always divide by the original value, not the new one (common trap)

Discounts, tax, tips, commission

Discount: pay (100% − discount%) of price; Tax/tip: pay (100% + rate%) of price
Multiply-by-remaining-percent shortcut: 25% off → ×0.75; 8% tax → ×1.08
Tax/tip applied to price after discount, not original price
Multiple discounts don’t add — apply sequentially (20% then 10% off ≠ 30% off)
To compare % discount vs. $ discount, convert % to dollars first
Splitting a tip: add tip to total first, then divide among people
Commission = percent × sales amount

Simple interest

Formula: $I = P \times r \times t$ (P=principal, r=decimal rate/year, t=years)
Watch for traps: one year’s interest only, confusing interest with final balance, or writing % wrong (4% ≠ 0.4)

Depreciation

Each year’s loss is calculated on the current value, not the original price
Multiply by remaining percent each year sequentially (e.g., ×0.80 repeatedly for 20% annual loss)
Don’t just subtract the same dollar amount each year

Rates and unit prices

Unit rate = total ÷ number of units (e.g., miles per gallon)
Unit price = cost ÷ units; lower unit price = better buy
Scale up by multiplying known rate by new quantity

Distance, rate, time

$distance = rate \times time$ ; rearrange for rate or time as needed
Units must match — convert minutes to hours (divide by 60) before using hourly rates
Multi-leg trips: compute time for each leg separately, don’t average speeds directly
True average speed = total distance ÷ total time (not simple average of two speeds)
Opposite directions: combined speed = sum of rates; same direction (chasing): relative speed = difference of rates

Work problems

Rates add when working simultaneously: combined rate = rate₁ + rate₂
Convert “time to complete alone” into a rate (fraction of job per hour) before adding
For machines/hours combo problems: find output per single unit per hour, then scale by new machines × new hours

Next | 4.3 Ratios, measurement, and probability

Percents, money, and rates

Percent questions are among the most common Arithmetic Reasoning problems. Nearly all of them come down to one relationship among three numbers: the part, the whole, and the percent.

The percent equation

$part = percent \times whole$

Write the percent as a decimal before you multiply: $35%$ is $0.35$ , $8%$ is $0.08$ , and $150%$ is $1.5$ .

Each question gives you two of the three numbers and asks for the third.

Finding the part: multiply. $15%$ of $60$ is $0.15 \times 60 = 9$ .
Finding the whole: divide the part by the percent. If $12$ is $30%$ of a number, the number is $12 \div 0.30 = 40$ .
Finding the percent: divide the part by the whole. $18$ out of $24$ is $18 \div 24 = 0.75$ , or $75%$ .

Percent increase and decrease

To find a percent change, divide the amount of change by the original amount:

$percent change = \frac{new - original}{original}$

A negative result is a decrease. A price that falls from $$100$ to $$80$ changes by $\frac{- 20}{100} = - 20%$ , so it decreased by $20%$ .

Example:

A worker’s hourly pay rises from $$16$ to $$20$ . By what percent did it increase?

A. $4%$
B. $20%$
C. $25%$
D. $80%$

The pay went up by $$20 - $16 = $4$ . Divide by the original pay:

$\frac{4}{16} = 0.25 = 25%$

Choice B divides by the new pay ( $4 \div 20$ ), which is the trap. A percent change is always measured from where the amount started.

Answer: C

Discounts, sales tax, tips, and commission

A discount comes off the price. Sales tax and a tip are added to it. Each is a percent of the price it’s figured on.

A quick way to handle either one is to multiply by what’s left. $25%$ off means you pay $75%$ of the price, so multiply by $0.75$ . An $8%$ tax means you pay $108%$ , so multiply by $1.08$ .

Example:

A pair of boots costs $$120$ . They’re on sale for $25%$ off, and sales tax is $5%$ . What is the total cost?

A. $$90.00$
B. $$94.50$
C. $$96.00$
D. $$126.00$

First, take off the discount: $$120 \times 0.75 = $90$ .

The tax is figured on the price you actually pay, which is $$90$ : $$90 \times 0.05 = $4.50$ .

Add it on: $$90 + $4.50 = $94.50$ .

Answer: B

Two discounts in a row don’t add. The second discount is taken off the already-reduced price. An item marked $20%$ off and then another $10%$ off isn’t $30%$ off:

$$200 \times 0.80 = $160, $160 \times 0.90 = $144$

The final price is $$144$ , not the $$140$ you’d get by taking $30%$ off.

To compare a percent discount with a dollar discount, turn the percent into dollars first. On a $$60$ item, $15%$ off is $0.15 \times $60 = $9$ off, which beats a $$8$ coupon by $$1$ .

To split a bill with a tip, add the tip first, then divide. A $$72$ bill with a $20%$ tip comes to $$72 \times 1.20 = $86.40$ , so $4$ friends each pay $$86.40 \div 4 = $21.60$ .

A commission is a percent of what a salesperson sells. At a $6%$ commission, $$4, 500$ in sales earns $0.06 \times $4, 500 = $270$ .

Simple interest

Simple interest is paid only on the original amount, called the principal:

$I = P \times r \times t$

Here $P$ is the principal, $r$ is the yearly interest rate written as a decimal, and $t$ is the time in years.

Example:

How much simple interest is earned on $$2, 000$ invested at $4%$ for $3$ years?

A. $$80$
B. $$240$
C. $$2, 240$
D. $$2, 400$

$I = 2, 000 \times 0.04 \times 3 = 240$

The other choices are common slips. $$80$ is one year of interest. $$2, 240$ is the account’s balance (principal plus interest), which answers a different question. $$2, 400$ comes from writing $4%$ as $0.4$ .

Answer: B

Depreciation

When something loses a percent of its value every year, each year’s loss is figured on what it was worth at the start of that year, not on the original price.

Example:

A $$1, 000$ machine loses $20%$ of its value each year. What is it worth after $2$ years?

Each year it keeps $80%$ of its value:

$$1, 000 \times 0.80 = $800 after one year$

$$800 \times 0.80 = $640 after two years$

Subtracting $$200$ twice gives $$600$ , which is wrong: in the second year the machine loses $20%$ of $$800$ , only $$160$ .

Answer: $$640$

Knowledge check

A jacket normally costs $$80$ . It’s on sale for $25%$ off. What is the sale price?

A. $$20$
B. $$55$
C. $$60$
D. $$65$

(spoiler)

$25%$ of $$80$ is $0.25 \times $80 = $20$ off the original price. Subtract the discount: $$80 - $20 = $60$ .

Answer: C. $$60$

Rates, distance, and work

A rate compares two different kinds of quantities: miles per hour, pages per minute, dollars per pound. “Per” means “for each one.”

Unit rates and unit prices

To find a unit rate, divide the total amount by the number of units.

Example:

A car travels $270$ miles on $9$ gallons of gas. How many miles per gallon does it get?

$270 \div 9 = 30$

Answer: $30$ miles per gallon

Once you know the rate, multiply to scale up. At $30$ miles per gallon, $12$ gallons take the car $30 \times 12 = 360$ miles.

A unit price is the cost of one unit, and it tells you which size is the better buy.

Example:

An $18$ -ounce box of cereal costs $$4.50$ , and a $24$ -ounce box costs $$5.52$ . Which is the better buy?

$$4.50 \div 18 = $0.25 per ounce$

$$5.52 \div 24 = $0.23 per ounce$

Answer: The $24$ -ounce box, which costs $2$ cents less per ounce.

Distance, rate, and time

The distance formula

$distance = rate \times time$

Rearranged: $time = distance \div rate$ , and $rate = distance \div time$ .

Example:

A cyclist rides for $2$ hours and $30$ minutes at a steady $14$ miles per hour. How far does the cyclist travel?

A. $28$ miles
B. $32.2$ miles
C. $35$ miles
D. $42$ miles

$2$ hours and $30$ minutes is $2.5$ hours, not $2.3$ hours.

$14 \times 2.5 = 35$

Answer: C

Averaging two speeds is a trap. When a trip has two legs at different speeds, find the time for each leg separately.

Example:

A driver travels $120$ miles at $60$ miles per hour, then another $120$ miles at $40$ miles per hour. What is the total travel time?

A. $4$ hours
B. $4.8$ hours
C. $5$ hours
D. $6$ hours

$120 \div 60 = 2 hours, 120 \div 40 = 3 hours$

The trip takes $2 + 3 = 5$ hours. Choice B averages the two speeds to $50$ miles per hour and divides $240$ miles by it. That’s wrong because the driver spends more time at the slower speed: the true average speed is $240 \div 5 = 48$ miles per hour.

Answer: C

Work problems

When two people or machines work at the same time, their rates add.

Example:

Printer A prints $20$ pages per minute, and Printer B prints $30$ pages per minute. Working at the same time, how long do they take to print $400$ pages?

Together they print $20 + 30 = 50$ pages per minute:

$400 \div 50 = 8$

Answer: $8$ minutes

When a problem changes both the number of machines and the hours, find what one machine does in one hour.

Example:

A factory makes $240$ parts in $4$ hours with $3$ machines running. At the same rate, how many parts can $5$ machines make in $6$ hours?

$240 \div 4 = 60$ parts per hour, from $3$ machines. That’s $60 \div 3 = 20$ parts per machine per hour.

$20 \times 5 \times 6 = 600$

Answer: $600$ parts

Key points

The percent equation

$part = percent \times whole$ ; convert percent to decimal first ( $35% = 0.35$ )
Finding part: multiply percent × whole
Finding whole: divide part by percent ( $12 \div 0.30 = 40$ )
Finding percent: divide part by whole, with whole (the “of” number) on bottom — dividing backwards is the common trap

Percent increase/decrease

$percent change = \frac{new - original}{original}$
Negative result = decrease
Always divide by the original value, not the new one (common trap)

Discounts, tax, tips, commission

Discount: pay (100% − discount%) of price; Tax/tip: pay (100% + rate%) of price
Multiply-by-remaining-percent shortcut: 25% off → ×0.75; 8% tax → ×1.08
Tax/tip applied to price after discount, not original price
Multiple discounts don’t add — apply sequentially (20% then 10% off ≠ 30% off)
To compare % discount vs. $ discount, convert % to dollars first
Splitting a tip: add tip to total first, then divide among people
Commission = percent × sales amount

Simple interest

Formula: $I = P \times r \times t$ (P=principal, r=decimal rate/year, t=years)
Watch for traps: one year’s interest only, confusing interest with final balance, or writing % wrong (4% ≠ 0.4)

Depreciation

Each year’s loss is calculated on the current value, not the original price
Multiply by remaining percent each year sequentially (e.g., ×0.80 repeatedly for 20% annual loss)
Don’t just subtract the same dollar amount each year

Rates and unit prices

Unit rate = total ÷ number of units (e.g., miles per gallon)
Unit price = cost ÷ units; lower unit price = better buy
Scale up by multiplying known rate by new quantity

Distance, rate, time

$distance = rate \times time$ ; rearrange for rate or time as needed
Units must match — convert minutes to hours (divide by 60) before using hourly rates
Multi-leg trips: compute time for each leg separately, don’t average speeds directly
True average speed = total distance ÷ total time (not simple average of two speeds)
Opposite directions: combined speed = sum of rates; same direction (chasing): relative speed = difference of rates

Work problems

Rates add when working simultaneously: combined rate = rate₁ + rate₂
Convert “time to complete alone” into a rate (fraction of job per hour) before adding
For machines/hours combo problems: find output per single unit per hour, then scale by new machines × new hours