Question

example 3 multiply mixed numbers
the canvas that morgan is using for her painting is shaped like a rectangle that is 1\frac{3}{4} yards long and 1\frac{1}{2} yards tall.
what is the area of the canvas? write the area as a mixed number in simplest form.
the canvas is 1\frac{3}{4} yards long and 1\frac{1}{2} yards tall. use the formula a = lw to find the area of the canvas.
a = lw \quad \text{area of a rectangle}
a = 1\frac{3}{4} \times 1\frac{1}{2} \quad \text{replace } l \text{ with } 1\frac{3}{4} \text{ and } w \text{ with } 1\frac{1}{2}.
a = \frac{7}{4} \times \frac{3}{2} \quad \text{rewrite the mixed numbers as fractions.}
a = \frac{7 \times 3}{4 \times 2} \quad \text{multiply the numerators and denominators.}
a = \frac{21}{8} \text{ or } 2\frac{5}{8} \quad \text{simplify}
so, the area of the canvas is \\_\\_\\_\\_\\_\\_\\_ square yards.

check
a recipe for a certain kind of bread calls for 2\frac{3}{4} cups of flour. juliet wants to make a batch of bread that is 1\frac{1}{2} times the recipe. how much flour does she need?

pause and reflect
in this lesson, you learned two different methods for multiplying fractions. how do you think these methods will help you with learning about dividing fractions?

Explanation:

First Problem (Canvas Area):

Step1: Recall the area formula for a rectangle.

The formula for the area \( A \) of a rectangle is \( A = l \times w \), where \( l \) is the length and \( w \) is the width.

Step2: Substitute the given mixed numbers for length and width.

The length \( l = 1\frac{3}{4} \) yards and the width \( w = 1\frac{1}{2} \) yards. So, \( A = 1\frac{3}{4} \times 1\frac{1}{2} \).

Step3: Convert mixed numbers to improper fractions.

To convert \( 1\frac{3}{4} \) to an improper fraction: \( 1\frac{3}{4}=\frac{1\times4 + 3}{4}=\frac{7}{4} \). To convert \( 1\frac{1}{2} \) to an improper fraction: \( 1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{3}{2} \). Now the equation becomes \( A=\frac{7}{4}\times\frac{3}{2} \).

Step4: Multiply the numerators and denominators.

When multiplying fractions, we multiply the numerators together and the denominators together. So, \( A = \frac{7\times3}{4\times2}=\frac{21}{8} \).

Step5: Convert the improper fraction to a mixed number.

To convert \( \frac{21}{8} \) to a mixed number, divide 21 by 8. \( 8\times2 = 16 \), and \( 21-16 = 5 \). So, \( \frac{21}{8}=2\frac{5}{8} \).

Step1: Identify the amount of flour in the recipe and the multiplier.

The recipe calls for \( 2\frac{3}{4} \) cups of flour, and Juliet wants to make \( 1\frac{1}{2} \) times the recipe. We need to find \( 2\frac{3}{4}\times1\frac{1}{2} \).

Step2: Convert mixed numbers to improper fractions.

Convert \( 2\frac{3}{4} \): \( 2\frac{3}{4}=\frac{2\times4 + 3}{4}=\frac{11}{4} \). Convert \( 1\frac{1}{2} \): \( 1\frac{1}{2}=\frac{1\times2 + 1}{2}=\frac{3}{2} \). Now the equation is \( \frac{11}{4}\times\frac{3}{2} \).

Step3: Multiply the numerators and denominators.

Multiply the numerators: \( 11\times3 = 33 \). Multiply the denominators: \( 4\times2 = 8 \). So, we get \( \frac{33}{8} \).

Step4: Convert the improper fraction to a mixed number.

Divide 33 by 8. \( 8\times4 = 32 \), and \( 33 - 32 = 1 \). So, \( \frac{33}{8}=4\frac{1}{8} \).

Answer:

The area of the canvas is \( 2\frac{5}{8} \) square yards.

Second Problem (Flour for Bread):