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Answer :
- To divide by a fraction, multiply by its reciprocal.
- The reciprocal of $\frac{5}{6}$ is $\frac{6}{5}$.
- Zachary should have multiplied $\frac{3}{4}$ by $\frac{6}{5}$.
- Therefore, Zachary should multiply $\frac{3}{4}$ by $\boxed{\frac{6}{5}}$.
### Explanation
1. Understanding the Problem
The problem is asking us to identify the correct number to multiply $\frac{3}{4}$ by in order to correctly perform the division $\frac{3}{4} \div \frac{5}{6}$. In Step 1, Zachary incorrectly wrote $\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}$. We need to determine what number Zachary should have multiplied $\frac{3}{4}$ by.
2. Finding the Correct Reciprocal
To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Therefore, to divide $\frac{3}{4}$ by $\frac{5}{6}$, we need to multiply $\frac{3}{4}$ by the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.
3. Identifying the Error
So, the correct expression should be $\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5}$. Comparing this to Zachary's incorrect Step 1, $\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}$, we can see that Zachary took the reciprocal of $\frac{3}{4}$ instead of $\frac{5}{6}$ and multiplied by $\frac{5}{6}$ instead of the reciprocal of $\frac{5}{6}$ which is $\frac{6}{5}$.
4. Final Answer
Therefore, Zachary should multiply $\frac{3}{4}$ by $\frac{6}{5}$ to fix his error in Step 1.
### Examples
When you're adjusting a recipe, you might need to divide a fraction of an ingredient by another fraction. For example, if you want to halve a recipe that calls for $\frac{3}{4}$ cup of flour, you're essentially dividing $\frac{3}{4}$ by 2 (or $\frac{2}{1}$). This is the same as multiplying $\frac{3}{4}$ by the reciprocal of 2, which is $\frac{1}{2}$. Understanding how to divide fractions is crucial for scaling recipes correctly, ensuring your dish turns out just right. This also applies to various other real-life scenarios, such as calculating distances on a map or determining proportions in construction projects.
- The reciprocal of $\frac{5}{6}$ is $\frac{6}{5}$.
- Zachary should have multiplied $\frac{3}{4}$ by $\frac{6}{5}$.
- Therefore, Zachary should multiply $\frac{3}{4}$ by $\boxed{\frac{6}{5}}$.
### Explanation
1. Understanding the Problem
The problem is asking us to identify the correct number to multiply $\frac{3}{4}$ by in order to correctly perform the division $\frac{3}{4} \div \frac{5}{6}$. In Step 1, Zachary incorrectly wrote $\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}$. We need to determine what number Zachary should have multiplied $\frac{3}{4}$ by.
2. Finding the Correct Reciprocal
To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Therefore, to divide $\frac{3}{4}$ by $\frac{5}{6}$, we need to multiply $\frac{3}{4}$ by the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.
3. Identifying the Error
So, the correct expression should be $\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5}$. Comparing this to Zachary's incorrect Step 1, $\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}$, we can see that Zachary took the reciprocal of $\frac{3}{4}$ instead of $\frac{5}{6}$ and multiplied by $\frac{5}{6}$ instead of the reciprocal of $\frac{5}{6}$ which is $\frac{6}{5}$.
4. Final Answer
Therefore, Zachary should multiply $\frac{3}{4}$ by $\frac{6}{5}$ to fix his error in Step 1.
### Examples
When you're adjusting a recipe, you might need to divide a fraction of an ingredient by another fraction. For example, if you want to halve a recipe that calls for $\frac{3}{4}$ cup of flour, you're essentially dividing $\frac{3}{4}$ by 2 (or $\frac{2}{1}$). This is the same as multiplying $\frac{3}{4}$ by the reciprocal of 2, which is $\frac{1}{2}$. Understanding how to divide fractions is crucial for scaling recipes correctly, ensuring your dish turns out just right. This also applies to various other real-life scenarios, such as calculating distances on a map or determining proportions in construction projects.
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