Math  /  Numbers & Operations

QuestionAre these ratios equivalent?
40 old necklaces : 35 new necklaces
16 old necklaces : 14 new necklaces yes no

Studdy Solution

STEP 1

1. We are given two ratios: 40 40 old necklaces to 35 35 new necklaces, and 16 16 old necklaces to 14 14 new necklaces.
2. We need to determine if these two ratios are equivalent.

STEP 2

1. Simplify the first ratio.
2. Simplify the second ratio.
3. Compare the simplified ratios.

STEP 3

Simplify the first ratio 40:35 40:35 .
Find the greatest common divisor (GCD) of 40 40 and 35 35 .

STEP 4

Calculate the GCD of 40 40 and 35 35 .
The factors of 40 40 are 1,2,4,5,8,10,20,40 1, 2, 4, 5, 8, 10, 20, 40 .
The factors of 35 35 are 1,5,7,35 1, 5, 7, 35 .
The GCD is 5 5 .

STEP 5

Divide both terms of the ratio 40:35 40:35 by their GCD, 5 5 .
405:355=8:7 \frac{40}{5} : \frac{35}{5} = 8 : 7

STEP 6

Simplify the second ratio 16:14 16:14 .
Find the greatest common divisor (GCD) of 16 16 and 14 14 .

STEP 7

Calculate the GCD of 16 16 and 14 14 .
The factors of 16 16 are 1,2,4,8,16 1, 2, 4, 8, 16 .
The factors of 14 14 are 1,2,7,14 1, 2, 7, 14 .
The GCD is 2 2 .

STEP 8

Divide both terms of the ratio 16:14 16:14 by their GCD, 2 2 .
162:142=8:7 \frac{16}{2} : \frac{14}{2} = 8 : 7

STEP 9

Compare the simplified ratios.
The simplified first ratio is 8:7 8:7 .
The simplified second ratio is 8:7 8:7 .
Since both simplified ratios are equal, the original ratios are equivalent.
The ratios are equivalent:
yes\boxed{\text{yes}}

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