Chapter 0.5, Problem 47E

To determine

The number of 9-inch ribbons that can be cut from $1\frac{1}{2}$ yards of ribbon.

Answer to Problem 47E

6 boards, each 2 feet 8 inches long, can be cut from a board 16 feet long.

Explanation of Solution

Given information:

A ribbon $1\frac{1}{2}$ yards long.

Formula used:

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

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

Conversion rule of converting yards into inches which can be mathematically expressed as,

  $\text{1 yard = 36 inches}$

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 a ribbon $1\frac{1}{2}$ yards long.

Recall that 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, $1\frac{1}{2}$ will be converted as,

  $\begin{array}{c}1\frac{1}{2}=\frac{\left(1\times 2\right)+1}{2}\\ =\frac{3}{2}\end{array}$

Recall conversion rule of converting yards into inches which can be mathematically expressed as,

  $\text{1 yard = 36 inches}$

So, $\frac{3}{2}$ yards will be evaluated as when $\frac{3}{2}$ is multiplied by $36$ .

  $\begin{array}{c}\text{1 yard = 36 inches}\\ \frac{\text{3}}{\text{2}}\text{yards =}\frac{\text{3}}{\text{2}}\cdot \text{36 inches}\\ \text{=54 inches}\end{array}$

Now, number of 9-inch ribbons that can be cut from $1\frac{1}{2}$ yards of ribbon, will be evaluated when $54$ is divided by $9$ .

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.

Sonumber of 9-inch ribbons that can be cut from $1\frac{1}{2}$ yards of ribbon, is calculated as,

  $\begin{array}{c}54÷9=54\cdot \frac{1}{9}\\ =\frac{9\cdot 6}{9}\\ =6\end{array}$

Thus, number of 9-inch ribbons that can be cut from $1\frac{1}{2}$ yards of ribbon is 6.