Math

Question Reduce the ratio 0.13:0.0520.13: 0.052 to its smallest terms.

Studdy Solution

STEP 1

Assumptions
1. The original ratio is 0.13:0.0520.13:0.052.
2. We need to reduce this ratio to its smallest terms.

STEP 2

To reduce the ratio to its smallest terms, we need to express both numbers as integers. To do this, we will find a common factor that can be used to eliminate the decimals.

STEP 3

We observe that both numbers can be multiplied by the same power of 10 to become integers. We will multiply both sides of the ratio by 10n10^n where nn is the number of decimal places in the number with the most decimal places.

STEP 4

In this case, both numbers have two decimal places, so we will multiply both sides of the ratio by 102=10010^2 = 100.
0.13×100:0.052×1000.13 \times 100 : 0.052 \times 100

STEP 5

Calculate the products to transform the decimals into integers.
13:5.213 : 5.2

STEP 6

Since we still have a decimal in the second number, we need to eliminate it by multiplying by an additional power of 10. Here, we multiply by 101=1010^1 = 10 to remove the decimal from 5.2.
13×10:5.2×1013 \times 10 : 5.2 \times 10

STEP 7

Calculate the products to get the integers.
130:52130 : 52

STEP 8

Now we have two integers. To reduce the ratio to its smallest terms, we need to find the greatest common divisor (GCD) of these two numbers.

STEP 9

Find the GCD of 130 and 52. We can do this by listing the factors of each number or by using the Euclidean algorithm.

STEP 10

The factors of 130 are: 1, 2, 5, 10, 13, 26, 65, 130. The factors of 52 are: 1, 2, 4, 13, 26, 52.

STEP 11

The greatest common factor that appears in both lists is 26.

STEP 12

Divide both numbers in the ratio by the GCD to reduce the ratio to its smallest terms.
13026:5226\frac{130}{26} : \frac{52}{26}

STEP 13

Calculate the division for both numbers.
5:25 : 2
The ratio in lowest terms is 5:25:2.

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