Questions: Making Change

Question 1

A pencil costs 23¢ and you pay with a quarter (25¢). Student A subtracts: 25 − 23 = 2, so 2¢ change. Student B counts up from 23¢: '24¢, 25¢' — two pennies. Which approach is correct?

• A. Only Student A — subtraction is the only correct method for making change
• B. Only Student B — counting up is the only method cashiers should use
• C. Both students are correct; subtraction and counting up are two strategies for the same calculation
• D. Neither — you need to count all the coins in your hand to find the change

Answer: (C)

Both methods work and reach the same answer (2¢) because they are two ways of computing the same thing: the difference between what was paid and what was owed. Subtraction computes it directly. Counting up reframes it as 'how much more is needed to get from 23¢ to 25¢?' Both are valid, and counting up is often faster with physical coins because you can stop as soon as you reach the amount paid.

Question 2

An item costs 37¢ and the customer pays with two quarters (50¢). Which combination gives correct change using the fewest coins?

• A. 13 pennies
• B. 1 dime and 3 pennies
• C. 2 nickels and 3 pennies
• D. 1 nickel and 8 pennies

Answer: (B)

Change is 50 − 37 = 13¢. The fewest coins: 1 dime (10¢) + 3 pennies (3¢) = 4 coins. Counting up from 37¢ naturally leads here: 'go to 40¢ (3 pennies), then to 50¢ (1 dime)' — 4 coins total. The 13-penny option uses the right total but the most coins. Using fewest coins is the practical goal because it is easier to count and verify, and it's what cashiers and vending machines aim for.

Question 3

The primary correct way to calculate change is to write out the subtraction problem (amount paid minus cost) before deciding which coins to return.

• T. True
• F. False

Answer: False

Counting up from the price to the amount paid is a completely valid — and often faster — strategy. Starting at the cost and adding coins until you reach the amount paid gives the exact same answer as subtraction, without needing paper or mental arithmetic. In real transactions, cashiers often count up by instinct: 'Your total is 83¢, you gave me a dollar — 84, 85, 90, one dollar' (2 pennies, 1 nickel, 1 dime = 17¢ change). The counting-up method also naturally produces the fewest coins.

Question 4

Change equals the amount paid minus the cost of the item.

• T. True
• F. False

Answer: True

This is the core equation: change = amount paid − cost. If an item costs 15¢ and you pay 25¢, change = 25 − 15 = 10¢. This is always true regardless of which calculation strategy you use. Counting up from 15¢ to 25¢ also gives 10¢ — it is the same calculation approached from a different direction (addition instead of subtraction), but the relationship is always amount paid − cost = change.

Question 5

How does counting up from the price help you give back change in fewer coins? Walk through the example of an item costing 47¢ when the customer pays with a dollar.

Model answer: Counting up naturally guides you toward friendly numbers (multiples of 5 and 10), which minimizes coins. Starting at 47¢: add 3 pennies to reach 50¢, then 2 quarters to reach $1.00. Total change: 3 pennies + 2 quarters = 53¢ in 5 coins. A direct subtraction gives $1.00 − $0.47 = 53¢, but doesn't tell you which coins to use. Counting up to the next 5¢ or 10¢ increment first naturally finds the efficient coin combination.

The counting-up strategy works because it exploits the structure of our coin system — values are at 1¢, 5¢, 10¢, 25¢, 50¢, $1.00. Moving to the nearest 'friendly' value (next multiple of 5 or 10) with small coins, then jumping with larger coins, minimizes the count. This is the same reasoning as rounding to a friendly number when estimating — you work toward round targets to keep the math simple and the coin count low.