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Pencils come in packages of 10 . Erasers come in packages of 12. Philip 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?

Pencils come in packages of $10$ . Erasers come in packages of $12$ . Philip 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?

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Solve a system of equations using any method: word problems

Full solution

Q. Pencils come in packages of

10

. Erasers come in packages of

12

. Philip 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?

Identify the problem: Identify the problem. $\newline$ Philip 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.
Determine the LCM: Determine the Least Common Multiple (LCM). $\newline$ To find the smallest number of pencils and erasers Philip can buy in whole packages, we need to find the LCM of the package sizes, which are $10$ and $12$ .
Calculate the 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 both numbers: $\text{LCM}(10, 12) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$ .
Determine packages needed: Determine the number of packages needed. $\newline$ Since the LCM is $60$ , Philip 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 the solution: Verify the solution. $\newline$ Philip should buy $6$ packages of pencils and $5$ packages of erasers. This will give him $60$ pencils and $60$ erasers, which is exactly $1$ eraser per pencil.

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