Question

write and solve a proportion to answer the question.
\\(\frac{3}{4}\\) is 60% of what number?
\\(\frac{\frac{3}{4}}{w} = \square\\)

Explanation:

Step1: Define the proportion

We know that 60% can be written as $\frac{60}{100}$ or $\frac{3}{5}$. The problem is asking " $\frac{3}{4}$ is 60% of what number? ", let the number be $n$. So the proportion should be $\frac{\frac{3}{4}}{n}=\frac{60}{100}$ (or $\frac{3}{5}$). Simplifying the right - hand side, $\frac{60}{100}=\frac{3}{5}$. So the proportion is $\frac{\frac{3}{4}}{n}=\frac{3}{5}$. Cross - multiplying gives us $3\times n=\frac{3}{4}\times5$. But also, we can think of it as $\frac{3}{4}$ is to $n$ as 60% (or $\frac{3}{5}$) is to 1. So $\frac{3}{4}\div n=\frac{3}{5}$, which is equivalent to $\frac{3}{4}=\frac{3}{5}\times n$. To solve for $n$, we can rewrite the proportion as $\frac{3}{4}=\frac{60}{100}\times n$, and then $n = \frac{\frac{3}{4}}{\frac{60}{100}}$.

Step2: Simplify the right - hand side

First, simplify $\frac{60}{100}=\frac{3}{5}$. Then, $\frac{\frac{3}{4}}{\frac{3}{5}}=\frac{3}{4}\times\frac{5}{3}$ (dividing by a fraction is multiplying by its reciprocal). The 3s cancel out, and we get $\frac{5}{4}$. So the proportion is $\frac{3}{4}=\frac{60}{100}\times n$ (or $\frac{3}{4}=\frac{3}{5}\times n$), and when we solve for the right - hand side of the given proportion $\frac{\frac{3}{4}}{n}=\frac{60}{100}$, the blank should be $\frac{60}{100}$ (or $\frac{3}{5}$). But let's solve for $n$ completely. From $\frac{3}{4}=0.6\times n$ (since 60% = 0.6), we can solve for $n$ by dividing both sides by 0.6. $n=\frac{\frac{3}{4}}{0.6}=\frac{3}{4}\div\frac{3}{5}=\frac{3}{4}\times\frac{5}{3}=\frac{5}{4} = 1.25$. But back to the proportion given in the problem: $\frac{\frac{3}{4}}{n}=\frac{60}{100}$, so the blank is $\frac{60}{100}$ (or $\frac{3}{5}$). If we want to write it as a fraction, $\frac{60}{100}$ simplifies to $\frac{3}{5}$.

Answer:

The blank should be filled with $\frac{60}{100}$ (or $\frac{3}{5}$). And the number $n=\frac{5}{4}$ (or 1.25). But for the proportion $\frac{\frac{3}{4}}{n}=\square$, the $\square$ is $\frac{60}{100}$ (or $\frac{3}{5}$).