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Arithmetic Reasoning

Ratios, Percents, Money & Rates

Many Arithmetic Reasoning questions describe everyday situations involving ratios, rates, prices, discounts, tax, tips, wages, overtime, commission, profit, and scaling quantities. The key is deciding what the numbers represent, what value a percent is based on, and whether a rate is used once or repeatedly.

Section 1

Ratios, Proportions & Scaling

A ratio compares two quantities. A proportion states that two ratios are equal. Keep a relationship constant by multiplying or dividing both quantities by the same factor.

For example, a mixture using 2 cups of concentrate for every 5 cups of water has the ratio 2:5. If every quantity is doubled, the ratio stays the same. Before setting up a proportion, check whether a simple multiplication or division is faster.

Example 1

A recipe uses 3 cups of flour to make 12 rolls. How many cups are needed to make 20 rolls?

Solution

  1. 1Set up a proportion: 312 = x20.
  2. 2Cross multiply: 3 × 20 = 12x.
  3. 360 = 12x, so divide by 12: x = 5.
Answer: 5 cups

Why: The amount of flour must increase in the same proportion as the number of rolls.

Example 2

A map uses a scale where 1 inch represents 25 miles. Two cities are 4 inches apart. What is the actual distance?

Solution

  1. 1Each inch represents 25 miles.
  2. 2Multiply for 4 inches: 4 × 25 = 100.
Answer: 100 miles

Why: The same rate of 25 miles applies to each inch of map distance.

Example 3

Red and blue paint are mixed in a ratio of 2:3. If 8 cups of red paint are used, how many cups of blue paint are needed?

Solution

  1. 1The red amount increases from 2 to 8, a scale factor of 4.
  2. 2Apply the same factor to blue: 3 × 4 = 12.
Answer: 12 cups of blue paint

Why: Both sides of a ratio must be multiplied by the same scale factor.

Section 2

Unit Rates & Comparing Values

A rate compares quantities with different units, such as miles per hour, dollars per pound, or pages per minute. A unit rate gives the amount for one unit.

To find a unit rate, divide by the number of units. To find a total from a known rate, multiply the rate by the number of units. For package prices, compare cost per item rather than the total price.

Example 1

A machine produces 420 parts in 7 hours. How many parts does it produce per hour?

Solution

  1. 1Find the amount for one hour: 420 ÷ 7 = 60.
Answer: 60 parts per hour

Why: A unit rate divides total output by the number of time units.

Example 2

Store A sells 15 batteries for $22.50. Store B sells 12 for $19.20. Which has the lower price per battery?

Solution

  1. 1Store A: 22.50 ÷ 15 = $1.50 per battery.
  2. 2Store B: 19.20 ÷ 12 = $1.60 per battery.
  3. 3Compare: $1.50 < $1.60.
Answer: Store A

Why: Comparing cost per battery puts packages of different sizes on the same basis.

Example 3

A worker packages 18 boxes per hour. How many boxes can the worker package in 7 hours?

Solution

  1. 1The hourly rate is repeated for 7 hours.
  2. 218 × 7 = 126.
Answer: 126 boxes

Why: Multiplication finds the total production from a known hourly rate.

Section 3

Percents, Discounts, Tax & Tips

Percent means per hundred. Convert a percent to a decimal, multiply by the correct base amount, and then decide whether the result is added or subtracted.

A discount is removed from the original price. Sales tax is added to the original price. A tip is an additional amount based on the bill and is added when the question asks for the total paid. Do not confuse a discount amount with the final sale price.

Example 1

A jacket costs $90 and is discounted by 30%. What is the sale price?

Solution

  1. 1Convert: 30% = 0.30.
  2. 2Find the discount: 0.30 × 90 = $27.
  3. 3Subtract: 90 - 27 = 63.
Answer: $63

Why: The 30% is the amount removed from the original price, not the final price.

Example 2

An item costs $75 and sales tax is 8%. What is the total cost?

Solution

  1. 1Convert: 8% = 0.08.
  2. 2Find tax: 0.08 × 75 = $6.
  3. 3Add tax: 75 + 6 = 81.
Answer: $81

Why: Sales tax increases the amount paid, so it is added to the original cost.

Example 3

A restaurant bill is $48 and the customer leaves a 20% tip. What is the total amount paid?

Solution

  1. 1Convert: 20% = 0.20.
  2. 2Find the tip: 0.20 × 48 = $9.60.
  3. 3Add it to the bill: 48 + 9.60 = 57.60.
Answer: $57.60

Why: The tip is calculated from the original bill and then added.

Section 4

Percent Increase, Decrease & Original Values

Percent change compares the amount changed with the ORIGINAL amount. Use percent change = change ÷ original × 100%.

For an increase, new - original is the change. For a decrease, original - new is the change. If a final value follows a discount, determine what percent remains before solving for the original amount.

Example 1

Weekly production rises from 200 units to 250 units. What is the percent increase?

Solution

  1. 1Find the increase: 250 - 200 = 50.
  2. 2Divide by the original: 50200 = 0.25.
  3. 3Convert: 0.25 = 25%.
Answer: 25%

Why: Percent increase measures change relative to where the amount started.

Example 2

A machine's operating cost falls from $500 to $425. What is the percent decrease?

Solution

  1. 1Find the decrease: 500 - 425 = 75.
  2. 2Divide by the original: 75500 = 0.15.
  3. 3Convert: 0.15 = 15%.
Answer: 15%

Why: The decrease is compared with the original $500, not the new $425.

Example 3

A television costs $680 after a 15% discount. What was its original price?

Solution

  1. 1A 15% discount means 85% remains.
  2. 2Convert: 85% = 0.85.
  3. 3Let the original price be x: 0.85x = 680.
  4. 4Divide: x = 800.
Answer: $800

Why: The $680 is 85% of the original because 15% was removed.

Section 5

Wages, Overtime & Commission

Separate each type of pay before combining it. Regular pay is hourly rate × hours. A common overtime rate is time-and-a-half, or 1.5 × the regular rate. Commission is a percent of sales.

If a worker has regular and overtime hours, find regular pay, find overtime pay, and then add them. If a salesperson receives base pay plus commission, calculate both separately and add them.

Example 1

A worker earns $19 per hour and works 32 hours. How much does the worker earn?

Solution

  1. 1Use hourly rate × hours.
  2. 219 × 32 = 608.
Answer: $608

Why: The same $19 rate is paid for each of the 32 hours.

Example 2

A worker earns $16 per hour, works 40 regular hours and 6 overtime hours, and overtime is time-and-a-half. What is total pay?

Solution

  1. 1Regular pay: 40 × 16 = $640.
  2. 2Overtime rate: 16 × 1.5 = $24 per hour.
  3. 3Overtime pay: 6 × 24 = $144.
  4. 4Add: 640 + 144 = 784.
Answer: $784

Why: Regular and overtime hours use different rates, so calculate each part separately.

Example 3

A salesperson earns $350 base pay plus 6% commission on $5,000 in sales. What is total pay?

Solution

  1. 1Commission: 0.06 × 5000 = $300.
  2. 2Add base pay: 350 + 300 = 650.
Answer: $650

Why: The salesperson receives both fixed base pay and percentage-based commission.

Section 6

Markup, Profit & Money Comparisons

Buying and selling problems distinguish cost, selling price, profit, loss, and markup. Compare the same quantity whenever possible and include every fee or charge.

Profit = selling price - cost. Loss = cost - selling price. A markup is a percentage of cost added to cost. Useful comparisons include price per item, cost per pound, cost per hour, total cost after fees, and total savings.

Example 1

A store buys an item for $45 and sells it for $68. What is the profit?

Solution

  1. 1Use profit = selling price - cost.
  2. 268 - 45 = 23.
Answer: $23

Why: Profit is what remains after subtracting the store's cost from its selling price.

Example 2

A store buys a table for $120 and marks it up by 30%. What is the selling price?

Solution

  1. 1Convert: 30% = 0.30.
  2. 2Find markup: 0.30 × 120 = $36.
  3. 3Add to cost: 120 + 36 = 156.
Answer: $156

Why: Markup is added to the original cost to determine the selling price.

Example 3

Plan A costs $25 plus $4 per use. Plan B costs $10 plus $6 per use. At 5 uses, which costs less?

Solution

  1. 1Plan A: 25 + 4(5) = 25 + 20 = $45.
  2. 2Plan B: 10 + 6(5) = 10 + 30 = $40.
  3. 3Compare: $40 < $45.
Answer: Plan B

Why: Each plan has a fixed cost and a per-use cost, so both parts must be included.

Quick Review

Ratios & Scaling

  • Compare quantities and multiply or divide both parts by the same factor.

Unit Rates

  • Divide to find an amount per one unit; multiply to find a total from a known rate.

Discounts, Tax & Tips

  • Find the percentage amount, then subtract a discount or add tax and tips when asked for the total.

Percent Change

  • Use change ÷ original × 100%; always compare with the original value.

Wages & Commission

  • Regular pay is rate × hours; overtime at time-and-a-half is 1.5 × the regular rate; add base pay to commission when given.

Profit & Markup

  • Profit is selling price - cost; markup is a percent of cost added to cost.

Arithmetic Reasoning Strategy

  • Identify quantities, find the base, decide multiply or divide, separate multiple steps, and check the result and units.