Chapter 0.5, Problem 45E

To determine

The number of band uniforms that can be made with $131\frac{3}{4}$ yards of fabric if each uniform requires $3\frac{7}{8}$ yards.

Answer to Problem 45E

  $34$ band uniforms can be made with $131\frac{3}{4}$ yards of fabric if each uniform requires $3\frac{7}{8}$ yards.

Explanation of Solution

Given information:

  $131\frac{3}{4}$ yards of fabric is to be used to make band uniforms of $3\frac{7}{8}$ yards .

  $$

Formula used:

Mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

Division in real numbers is multiplication by reciprocal. If $b\ne 0$ , then,

  $\begin{array}{c}a÷b=a\left(\frac{1}{b}\right)\\ =\frac{a}{b}\end{array}$

Here, $\frac{1}{b}$ is the multiplicative inverse.

Calculation:

Consider the given statement “ $131\frac{3}{4}$ yards of fabric is to be used to make band uniforms of $3\frac{7}{8}$ yards.”

Recall that the mixed fractions of the form $m\frac{p}{q}$ can be converted into improper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

So, $131\frac{3}{4}$ will be converted as,

  $\begin{array}{c}131\frac{3}{4}=\frac{\left(131\times 4\right)+3}{4}\\ =\frac{527}{4}\end{array}$

And, $3\frac{7}{8}$ will be converted as,

  $\begin{array}{c}3\frac{7}{8}=\frac{\left(3\times 8\right)+7}{8}\\ =\frac{31}{8}\end{array}$

Now, the number of band uniforms that can be made with $131\frac{3}{4}$ yards of fabric if each uniform requires $3\frac{7}{8}$ yards will be evaluated when $131\frac{3}{4}$ is divided by $3\frac{7}{8}$ .

Recall that the division in real numbers is multiplication by reciprocal. If $b\ne 0$ , then,

  $\begin{array}{c}a÷b=a\left(\frac{1}{b}\right)\\ =\frac{a}{b}\end{array}$

Here, $\frac{1}{b}$ is the multiplicative inverse.

So, number of band uniforms that can be made with $131\frac{3}{4}$ yards of fabric if each uniform requires $3\frac{7}{8}$ yards is calculated as,

  $\begin{array}{c}131\frac{3}{4}÷3\frac{7}{8}=\frac{527}{4}÷\frac{31}{8}\\ =\frac{527}{4}\cdot \frac{8}{31}\end{array}$

8 can be expressed as a product of 4 and 2.

  $527$ can be expressed as a product of $31$ and $17$ .

So, it can be further simplified as,

  $\begin{array}{c}131\frac{3}{4}÷3\frac{7}{8}=\frac{527}{4}\cdot \frac{8}{31}\\ =\frac{31\cdot 17\cdot 4\cdot 2}{4\cdot 31}\end{array}$

Striking off the common terms, we get,

  $\begin{array}{c}131\frac{3}{4}÷3\frac{7}{8}=\frac{31\cdot 17\cdot 4\cdot 2}{4\cdot 31}\\ =17\cdot 2\\ =34\end{array}$

Thus, $34$ band uniforms can be made with $131\frac{3}{4}$ yards of fabric if each uniform requires $3\frac{7}{8}$ yards.