Write [tex]$\frac{7}{5}+\frac{2}{3 r}$[/tex] as a single fraction in its simplest form.
Answer (2)
To add the fractions, find the least common denominator, rewrite each fraction with the common denominator, add the numerators, and simplify the result.
Find the least common denominator: 15 r .
Rewrite the fractions: 5 7 = 15 r 21 r and 3 r 2 = 15 r 10 .
Add the fractions: 15 r 21 r + 15 r 10 = 15 r 21 r + 10 .
The simplified form is: 15 r 21 r + 10 .
Explanation
Problem Analysis We want to combine two fractions, 5 7 and 3 r 2 , into a single fraction in its simplest form.
Finding the Least Common Denominator To add fractions, we need a common denominator. The least common denominator (LCD) of 5 and 3 r is 15 r .
Rewriting Fractions with Common Denominator Now, we rewrite each fraction with the LCD of 15 r :
For the first fraction: 5 7 = 5 7 × 3 r 3 r = 5 × 3 r 7 × 3 r = 15 r 21 r
For the second fraction: 3 r 2 = 3 r 2 × 5 5 = 3 r × 5 2 × 5 = 15 r 10
Adding the Fractions Now that both fractions have the same denominator, we can add them: 15 r 21 r + 15 r 10 = 15 r 21 r + 10
Simplifying the Result The numerator is 21 r + 10 and the denominator is 15 r . We need to check if this fraction can be simplified. Since there are no common factors between 21 r + 10 and 15 r , the fraction is already in its simplest form.
Examples
When combining different ingredients in a recipe, you often need to add fractional amounts. For example, if a recipe calls for 5 7 cups of flour and 3 r 2 cups of sugar (where r depends on the number of servings), expressing the total amount of dry ingredients as a single fraction helps in measuring the correct proportions. This ensures the recipe turns out as expected, maintaining the right balance of flavors and textures.
To add the fractions 5 7 and 3 r 2 , we find the least common denominator, which is 15 r . After rewriting both fractions, we combine them to get 15 r 21 r + 10 , which is in its simplest form. No further simplification is possible as there are no common factors in the numerator and denominator.
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