Simplify [tex]$\frac{10}{16}$[/tex] to lowest terms and find an equivalent fraction that has a denominator of 32.
A. [tex]$\frac{6}{16}, \frac{20}{32}$[/tex]
B. [tex]$\frac{20}{32}, \frac{10}{16}$[/tex]
C. [tex]$\frac{5}{8}, \frac{20}{32}$[/tex]
D. [tex]$\frac{5}{8}, \frac{10}{32}$[/tex]
Answer (2)
Simplify the fraction 16 10 by finding the greatest common divisor (GCD) of 10 and 16, which is 2, and dividing both the numerator and the denominator by 2: 16 ÷ 2 10 ÷ 2 = 8 5 .
Find the multiplier to convert the denominator 16 to 32 by dividing 32 by 16: 32 ÷ 16 = 2 .
Multiply both the numerator and the denominator of 16 10 by 2 to find the equivalent fraction with a denominator of 32: 16 × 2 10 × 2 = 32 20 .
The simplified fraction is 8 5 , and the equivalent fraction is 32 20 , so the answer is 8 5 , 32 20 .
Explanation
Problem Analysis We need to simplify the fraction 16 10 to its lowest terms and find an equivalent fraction with a denominator of 32.
Finding the GCD To simplify 16 10 , we need to find the greatest common divisor (GCD) of 10 and 16. The GCD is the largest number that divides both 10 and 16 without leaving a remainder.
Listing Factors The factors of 10 are 1, 2, 5, and 10. The factors of 16 are 1, 2, 4, 8, and 16. The greatest common divisor of 10 and 16 is 2.
Simplifying the Fraction Now, we divide both the numerator and the denominator of 16 10 by their GCD, which is 2: 16 ÷ 2 10 ÷ 2 = 8 5 So, the simplified fraction is 8 5 .
Finding Equivalent Fraction Next, we need to find an equivalent fraction to 16 10 with a denominator of 32. To do this, we need to find a number that we can multiply the denominator 16 by to get 32.
Calculating Equivalent Fraction We can find this number by dividing 32 by 16: 32 ÷ 16 = 2 So, we need to multiply both the numerator and the denominator of 16 10 by 2 to get the equivalent fraction with a denominator of 32: 16 × 2 10 × 2 = 32 20
Final Answer Therefore, the simplified fraction is 8 5 , and the equivalent fraction with a denominator of 32 is 32 20 .
Examples
Imagine you are baking a cake and the recipe calls for 16 10 of a cup of flour. To make it easier to measure, you simplify the fraction to 8 5 of a cup. Later, you decide to double the recipe. Now you need to use an equivalent fraction with a denominator of 32, which is 32 20 of a cup. This example shows how simplifying fractions and finding equivalent fractions can be useful in everyday situations like cooking.
The simplified fraction of 16 10 is 8 5 and the equivalent fraction with a denominator of 32 is 32 20 . The correct option is C: 8 5 , 32 20 .
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