Write $\frac{3}{4 r}+\frac{2}{p}$ 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. The least common denominator of 4 r and p is 4 r p . Rewrite the fractions: 4 r 3 = 4 r p 3 p and p 2 = 4 r p 8 r . Add the fractions: 4 r p 3 p + 4 r p 8 r = 4 r p 3 p + 8 r . The final answer is 4 r p 3 p + 8 r .
Explanation
Problem Analysis We are asked to write the sum of two fractions, 4 r 3 and p 2 , as a single fraction in its simplest form. To do this, we need to find a common denominator for the two fractions and then add them together.
Finding the Least Common Denominator The least common denominator (LCD) of 4 r and p is 4 r p , since 4 r and p do not share any common factors.
Rewriting Fractions with the LCD Now, we rewrite each fraction with the LCD as the denominator: 4 r 3 = 4 r 3 ⋅ p p = 4 r p 3 p p 2 = p 2 ⋅ 4 r 4 r = 4 r p 8 r
Adding the Fractions Next, we add the two fractions: 4 r p 3 p + 4 r p 8 r = 4 r p 3 p + 8 r
Simplifying the Result Finally, we check if the resulting fraction can be simplified. In this case, 3 p + 8 r and 4 r p do not have any common factors in general, so the fraction is already in its simplest form.
Examples
Fractions are a fundamental concept in mathematics and are used in various real-life situations. For instance, when you're cooking and need to combine different ingredients in specific ratios, you're working with fractions. If a recipe calls for 4 3 cup of flour and 3 2 cup of sugar, you need to find a common denominator to understand the total amount of dry ingredients you're adding. This skill is also useful in managing finances, calculating distances, or understanding proportions in science and engineering.
To combine 4 r 3 + p 2 into a single fraction, we find the least common denominator 4 r p , rewrite each fraction accordingly, and then add them. The result is 4 r p 3 p + 8 r , which is in its simplest form. Therefore, the final answer is 4 r p 3 p + 8 r .
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