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Solve each word problem by finding GCF or LCM.
Pencils come in packages of 10. Erasers come in packages of 12. Phillip wants to purchase the smallest number of pencils and erasers so that he will have exactly 1 eraser per pencil. How many packages of pencils and erasers should Phillip buy?
A. 4 packages of pencils and 3 packages of erasers
B. 5 packages of pencils and 4 packages of erasers
C. 6 packages of pencils and 5 packages of erasers
D. 12 packages of pencils and 10 packages of erasers

Solve each word problem by finding GCF or LCM. $\newline$ Pencils come in packages of $10$ . Erasers come in packages of $12$ . Phillip wants to purchase the smallest number of pencils and erasers so that he will have exactly $1$ eraser per pencil. How many packages of pencils and erasers should Phillip buy? $\newline$ A. $4$ packages of pencils and $3$ packages of erasers $\newline$ B. $5$ packages of pencils and $4$ packages of erasers $\newline$ C. $6$ packages of pencils and $5$ packages of erasers $\newline$ D. $12$ packages of pencils and $10$ packages of erasers

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Solve a system of equations using any method: word problems

Full solution

Q. Solve each word problem by finding GCF or LCM.

\newline

Pencils come in packages of

10

. Erasers come in packages of

12

. Phillip wants to purchase the smallest number of pencils and erasers so that he will have exactly

1

eraser per pencil. How many packages of pencils and erasers should Phillip buy?

\newline

A.

4

packages of pencils and

3

packages of erasers

\newline

B.

5

packages of pencils and

4

packages of erasers

\newline

C.

6

packages of pencils and

5

packages of erasers

\newline

D.

12

packages of pencils and

10

packages of erasers

Understand the problem: Understand the problem. $\newline$ Phillip wants to have exactly $1$ eraser for each pencil. Pencils come in packages of $10$ and erasers come in packages of $12$ . We need to find the smallest number of packages he should buy to have the same number of pencils and erasers.
Identify concept: Identify the mathematical concept needed to solve the problem. $\newline$ To ensure Phillip has the same number of pencils and erasers, we need to find the least common multiple (LCM) of the package sizes for pencils ( $10$ ) and erasers ( $12$ ).
Calculate LCM: Calculate the LCM of $10$ and $12$ . The prime factorization of $10$ is $2 \times 5$ . The prime factorization of $12$ is $2^2 \times 3$ . The LCM is the product of the highest powers of all prime factors present in the numbers, so $\text{LCM}(10, 12) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$ .
Determine packages needed: Determine the number of packages needed. $\newline$ Since the LCM is $60$ , Phillip needs to buy enough packages to have $60$ pencils and $60$ erasers. $\newline$ For pencils, $60$ pencils $/$ $10$ pencils per package $=$ $6$ packages. $\newline$ For erasers, $60$ erasers $/$ $60$ $0$ erasers per package $=$ $60$ $2$ packages.
Verify solution: Verify the solution. $\newline$ Phillip needs $6$ packages of pencils and $5$ packages of erasers to have exactly $1$ eraser per pencil. This matches one of the provided options, confirming that the calculations are correct.

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