Question

stuart bought a candy bar that cost 90 cents. he gave the cashier a $1 bill. which of the following tree diagrams shows all of the ways the cashier can give correct change using pennies, nickels, and dimes? 0 nickels - 10 pennies 0 dimes 1 nickel - 5 pennies 1 dime - 0 nickels - 0 pennies 0 dimes - 0 nickels - 0 pennies 0 nickels - 10 pennies 1 dime 1 nickel - 5 pennies 2 nickels - 0 pennies 0 nickels - 10 pennies 0 dimes 1 nickel - 5 pennies 2 nickels - 0 pennies 1 dime - 0 nickels - 0 pennies 1 nickel - 5 pennies 0 dimes 2 nickels - 0 pennies 1 dime - 0 nickels - 0 pennies

Explanation:

Step1: Calculate the change amount

Stuart gave $1$ (which is $100$ cents) for a $90$ - cent candy bar. The change is $100 - 90=10$ cents.

Step2: Analyze the first tree - diagram option

• If there are $0$ dimes:
• If $0$ nickels, then the number of pennies is $10$ (since $0\times10 + 0\times5+10\times1 = 10$).
• If $1$ nickel, then the number of pennies is $5$ (since $0\times10+1\times5 + 5\times1=10$).
• If there is $1$ dime ($10$ cents), then $0$ nickels and $0$ pennies (since $1\times10+0\times5 + 0\times1=10$).

Step3: Analyze the second tree - diagram option

When there are $0$ dimes, having $0$ nickels and $0$ pennies gives $0$ cents, which is not part of the $10$ - cent change. So this option is incorrect.

Step4: Analyze the third tree - diagram option

When there is $1$ dime, we don't need to consider multiple nickel - penny combinations for the dime case. The change is already accounted for by the dime alone. So this option is incorrect.

Step5: Analyze the fourth tree - diagram option

Similar to the third option, when there is $1$ dime, we don't need to consider nickel - penny combinations for the dime case. Also, the $0$ - dime case is not fully represented (missing the $0$ - nickel, $10$ - penny combination). So this option is incorrect.

Answer:

The first tree - diagram (the one with \(0\) dimes having \(0\) nickels - \(10\) pennies and \(1\) nickel - \(5\) pennies, and \(1\) dime having \(0\) nickels - \(0\) pennies) shows all the ways the cashier can give correct change.