Multiplying and Dividing Fractions

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Questions and Answers

Explain how you would divide $3$ by $\frac{2}{5}$, showing each step.

First, write $3$ as $\frac{3}{1}$. Then, to divide by $\frac{2}{5}$, multiply by its reciprocal $\frac{5}{2}$. So, $\frac{3}{1} \times \frac{5}{2} = \frac{15}{2} = 7\frac{1}{2}$.

Describe the steps to simplify the fraction $\frac{18}{24}$ to its lowest terms.

Find the greatest common factor (GCF) of 18 and 24, which is 6. Then, divide both the numerator and the denominator by the GCF: $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$.

What is the reciprocal of $2\frac{1}{4}$, and how did you find it?

First, convert $2\frac{1}{4}$ to an improper fraction: $2\frac{1}{4} = \frac{9}{4}$. The reciprocal is then $\frac{4}{9}$.

Explain how the concept of reciprocals is used when dividing one fraction by another.

Dividing by a fraction is the same as multiplying by its reciprocal. Instead of dividing by the second fraction, you flip it (find its reciprocal) and then multiply. Signup and view all the answers

Provide an example of a real-world scenario where dividing fractions might be necessary.

If a recipe calls for $\frac{2}{3}$ cup of flour and you only want to make half the recipe, you would need to calculate $\frac{2}{3} \div 2$, which means dividing a fraction. Signup and view all the answers

Solve: $\frac{5}{8} \div \frac{2}{3} = $?

$\frac{5}{8} \times \frac{3}{2} = \frac{15}{16}$ Signup and view all the answers

Solve: $\frac{9}{5} \div 3 = $?

$\frac{9}{5} \times \frac{1}{3} = \frac{9}{15} = \frac{3}{5}$ Signup and view all the answers

How do you find the GCF, and why is it an important step when simplifying fractions?

The GCF is the greatest number that divides evenly into both the numerator and denominator. It's important because dividing by the GCF ensures the fraction is simplified to its lowest terms. Signup and view all the answers

Explain how to divide a mixed number by a fraction, using $2\frac{1}{2} \div \frac{3}{4}$ as an example.

First, convert $2\frac{1}{2}$ to an improper fraction: $\frac{5}{2}$. Then, divide by $\frac{3}{4}$ by multiplying by its reciprocal $\frac{4}{3}$. Thus $\frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$. Signup and view all the answers

What is the result of any non-zero number multiplied by its reciprocal? Give an example.

The result is always 1. For example, $\frac{4}{7} \times \frac{7}{4} = 1$. Signup and view all the answers

Explain why multiplying a fraction by its reciprocal always results in 1.

When you multiply a fraction by its reciprocal, you are essentially multiplying the numerator by the original denominator and the denominator by the original numerator. This results in the same number in both the numerator and denominator, which simplifies to 1. Signup and view all the answers

Describe a situation where simplifying fractions before multiplying is particularly useful and why.

Simplifying before multiplying is particularly useful when dealing with larger numbers in the numerators and denominators. It reduces the size of the numbers, making the multiplication and final simplification easier and less prone to error. Signup and view all the answers

How does multiplying a fraction by a number greater than 1 affect the fraction's value? What about multiplying by a number between 0 and 1?

Multiplying a fraction by a number greater than 1 increases the fraction's value. Multiplying by a number between 0 and 1 decreases the fraction's value. Signup and view all the answers

If you divide a fraction by 2, is the result the same as multiplying it by 1/2? Explain why or why not.

Yes, dividing a fraction by 2 is the same as multiplying by 1/2 because dividing by a number is the same as multiplying by its reciprocal, and the reciprocal of 2 is 1/2. Signup and view all the answers

Explain the difference in approach when adding fractions versus multiplying them. Why don't we need a common denominator when multiplying?

When adding fractions, a common denominator is needed to ensure that we are adding like terms (parts of the same whole). When multiplying, we are finding a fraction of a fraction, so the denominators do not need to be the same; we simply multiply straight across. Signup and view all the answers

Describe the steps to solve the following: $2\frac{1}{3} \div \frac{3}{4}$

First, convert the mixed number to an improper fraction: $2\frac{1}{3} = \frac{7}{3}$. Then, multiply by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$. So, $\frac{7}{3} \times \frac{4}{3} = \frac{28}{9}$. Finally, convert back to mixed number form to get $3\frac{1}{9}$. Signup and view all the answers

During a bake-off, you need to halve a recipe that calls for $\frac{2}{3}$ cup of sugar. How much sugar do you need? Show the math.

Halving a recipe means multiplying by $\frac{1}{2}$. Therefore, $\frac{2}{3} \times \frac{1}{2} = \frac{2}{6}$, which simplifies to $\frac{1}{3}$ cup of sugar. Signup and view all the answers

You have $\frac{3}{4}$ of a pizza left, and you want to share it equally among 3 friends. What fraction of the whole pizza does each friend get?

To share equally, divide the fraction of the pizza by the number of friends: $\frac{3}{4} \div 3 = \frac{3}{4} \div \frac{3}{1} = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12}$, which simplifies to $\frac{1}{4}$. Each friend gets $\frac{1}{4}$ of the whole pizza. Signup and view all the answers

If a cyclist covers $\frac{2}{5}$ of a race distance in one hour, how much of the distance would they cover in $2\frac{1}{2}$ hours, assuming a constant speed?

First, convert the mixed number to an improper fraction: $2\frac{1}{2} = \frac{5}{2}$. Then, multiply the fraction of the distance covered in one hour by the number of hours: $\frac{2}{5} \times \frac{5}{2} = \frac{10}{10}$, which simplifies to 1. The cyclist would cover the entire distance in $2\frac{1}{2}$ hours. Signup and view all the answers

A gardener plants $\frac{1}{3}$ of a garden with flowers and divides the remaining space equally between vegetables and herbs. What fraction of the total garden is used for herbs?

First find the fraction of the garden remaining after planting flowers: $1 - \frac{1}{3} = \frac{2}{3}$. Then, divide the remaining space between vegetables and herbs: $\frac{2}{3} \div 2 = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6}$, which simplifies to $\frac{1}{3}$. Therefore, $\frac{1}{3}$ of the total garden is used for herbs. Signup and view all the answers

Flashcards

Multiplying Fractions

Multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator: (a/b) * (c/d) = (ac)/(bd).

Simplifying Fractions Before Multiplying

Cancel common factors between any numerator and any denominator before multiplying.