Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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?

View step-by-step help

View step-by-step help

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.

More problems from Solve a system of equations using any method: word problems

Question

Solve using elimination.

\newline

x + 6y = -15

\newline

-2x + 6y = 12

\newline

(\_,\_)

Posted 2 months ago

Question

(1,1)

a solution to this system of equations?

\newline

8x + y = 9

\newline

6x + 12y = 18

\newline

\newline

\newline

Posted 1 month ago

Question

How many solutions does the system of equations below have?

\newline

7

\newline

7

\newline

\newline

\text{[A]no solution}

\newline

\text{[B]one solution}

\newline

\text{[C]infinitely many solutions}

Posted 1 month ago

Question

Which describes the system of equations below?

\newline

y = \frac{1}{7}x - 10

\newline

y = \frac{1}{7}x - 10

\newline

\newline

\text{[A]consistent and independent}

\newline

\text{[B]consistent and dependent}

\newline

\text{[C]inconsistent}

Posted 1 month ago

Question

Solve using substitution.

\newline

y = -5

\newline

-4x + 3y = 17

\newline

Posted 1 month ago

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Last Wednesday, two friends met up after school to read the book they were both assigned in Literature class. Bryan can read

2

pages per minute, and he had already read

98

pages. Jayla, who has a reading speed of

3

pages per minute, had read

48

pages. Eventually they had read the same number of pages. How long did that take? How many pages had each of them read?

\newline

\_\_\_\_

minutes, Bryan and Jayla had each read

\_\_\_\_

Posted 1 month ago

Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

At the Castroville Ice Cream Parlor, one group of friends ordered

1

small serving of ice cream and

3

large servings of ice cream for

20

. Another group of friends ordered

1

small serving of ice cream and

4

large servings of ice cream for

26

. How much does the ice cream cost?

\newline

The ice cream costs `\$`_____ for a small serving and `\$`_____ for a large serving.

Posted 1 month ago

Question

\newline

y = -1

\newline

-2x - 4y = 8

\newline

(_____ , _____)

Posted 1 month ago

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

There are two mystery numbers. The sum of

3

times the first number and

3

times the second number is

-18

2

times the first number and

3

times the second number is

-13

. What are the two numbers?

\newline

The first number is ______ , and the second number is ______.

Posted 1 month ago

Question

A teacher purchased

300

colored pencils for an upcoming project. The pencils she ordered come in packs of

20

30

14

packs in all. Write a system of equations to find the number of

20

(x)

30

(y)

the art teacher ordered.

Posted 2 months ago