Examkin

Question

Difficulty: MediumFactors, Multiples, and Prime Factorization

An artist has a rectangular sheet of stained glass that measures 8484 inches by 120120 inches. The artist wants to cut the sheet into congruent square tiles of the largest possible side length, such that there is no glass wasted. What is the total number of square tiles the artist will obtain?

Answer: 70

Answer

The total number of square tiles the artist will obtain is 70.
To find the maximum side length of the square tiles without wasting any glass, we must find the greatest common factor (GCF) of the rectangular dimensions, 8484 and 120120. The prime factorizations are 84=22×3×784 = 2^2 \times 3 \times 7 and 120=23×3×5120 = 2^3 \times 3 \times 5, which gives a GCF of 22×3=122^2 \times 3 = 12 inches. The number of square tiles that fit along the length is 84÷12=784 \div 12 = 7 and along the width is 120÷12=10120 \div 12 = 10. Multiplying these dimensions gives a total of 7×10=707 \times 10 = 70 square tiles.

Step-by-Step Solution

1
Determine the prime factorizations of the dimensions of the sheet.
84=22×3×784 = 2^2 \times 3 \times 7 and 120=23×3×5120 = 2^3 \times 3 \times 5
This allows finding the greatest common factor of the two side lengths.
2
Calculate the greatest common factor (GCF) of 8484 and 120120.
GCF(84,120)=22×3=12\text{GCF}(84, 120) = 2^2 \times 3 = 12
The largest congruent squares that can tile the sheet without waste must have a side length equal to this GCF.
3
Divide each dimension of the sheet by the tile side length to find the number of tiles along each side.
84÷12=784 \div 12 = 7 tiles along the width, and 120÷12=10120 \div 12 = 10 tiles along the length.
This determines the grid dimensions of the tiles.
4
Multiply the number of tiles along the width by the number of tiles along the length.
7×10=707 \times 10 = 70
The total number of square tiles is the product of the number of tiles along each dimension.

Key Concept

Greatest Common Factor (GCF) application to division of a two-dimensional grid
Rate this question

Topics

Factors, Multiples, and Prime Factorization

More questions from this topic