Chapter 2.2, Problem 83PFA

(a)

To determine

Write an equation to represent the situation.

Answer to Problem 83PFA

  $\frac{3}{8}·t=120$

Explanation of Solution

Given:

At a camp site in July, 120 families stayed for three or more days. The 120 families represent three − eighths of all the families who stayed at the campsite in July.

Concept Used:

The 120 families represent three − eighths of all the families who stayed at the campsite in July.

Let t = the total number of families

Three − eighths of all the families = $\frac{3}{8}·t$

The 120 families represent three − eighths of all the families: $\frac{3}{8}·t=120$

Calculation:

Equation: $\frac{3}{8}·t=120$

Thus, the equation represent the situation is $\frac{3}{8}·t=120$

(b)

To determine

Find the number of families stayed at the campsite.

Answer to Problem 83PFA

Option C is correct.

Explanation of Solution

Given:

At a camp site in July, 120 families stayed for three or more days. The 120 families represent three − eighths of all the families who stayed at the campsite in July.

Concept Used:

The 120 families represent three − eighths of all the families who stayed at the campsite in July.

Let t = the total number of families

Three − eighths of all the families = $\frac{3}{8}·t$

The 120 families represent three − eighths of all the families: $\frac{3}{8}·t=120$

Calculation:

Equation: $\frac{3}{8}·t=120$

Solve fort = total number of family member.

  $\begin{array}{l}\frac{3}{8}·t=120\\ \frac{8}{3}·\frac{3}{8}·t=\frac{8}{3}·120\text{[}\text{Multiply}\text{both}\text{sides}\text{by}\text{the}\text{reciprocal}\text{of}\text{the}\text{coefficent}\text{of}t\text{]}\\ \text{t}\text{=}\frac{960}{3}[\frac{8}{3}·\frac{3}{8}=1]\\ t=320\end{array}$

Total number of family member is 320. Option C is correct.

Thus, total number of family member is 320. Option C is correct.

(c)

To determine

Write an equation to represent this situation.

Answer to Problem 83PFA

  $40\%·t=120$

Explanation of Solution

Given:

In August, 120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Let t = the total number of families.

Concept Used:

Let t = the total number of families.

120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Convert the condition given in the question into algebraic equation.

Let t = the total number of families.

40% of all the families = $40\%·t$

Total number of families stayed in August = 120 families.

Equation: $40\%·t=120$

Calculation:

Equation: $40\%·t=120$

Thus, the equation represent the situation is $40\%·t=120$

(d)

To determine

Find the number of families stayed at the campsite in August?

Answer to Problem 83PFA

300

Explanation of Solution

Given:

In August, 120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Let t = the total number of families.

Concept Used:

120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Convert the condition given in the question into algebraic equation.

Let t = the total number of families.

40% of all the families = $40\%·t$

Total number of families stayed in August = 120 families.

Equation: $40\%·t=120$

Calculation:

Equation: $40\%·t=120$

Solve for t = the total number of families.

The coefficient of t is $40\%\text{or}\frac{40}{100}$

The reciprocal of the coefficient of t is $\frac{40}{100}$ and $\frac{100}{40}·\frac{40}{100}=1$

  $\begin{array}{l}\frac{40}{100}·t=120\\ \frac{100}{40}·\frac{40}{100}·t=\frac{100}{40}·120[\text{Multiply}\text{both}\text{sides}\text{by}\text{the}\text{reciprocal}\text{of}\text{the}\text{coefficient}\text{of}t]\\ t=\frac{12000}{40}[\frac{100}{40}·\frac{40}{100}=1]\\ t=300\end{array}$

The total number of families stayed in August = 300

Thus, the total number of families in August is 300.

(e)

To determine

Describe how you could solve the equation from part C by using multiplication.

Answer to Problem 83PFA

Multiply each side by the reciprocal of the coefficient of t

Explanation of Solution

Given:

In August, 120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Let t = the total number of families.

Concept Used:

120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Convert the condition given in the question into algebraic equation.

Let t = the total number of families.

40% of all the families = $40\%·t$

Total number of families stayed in August = 120 families.

Equation: $40\%·t=120$

Calculation:

Equation: $40\%·t=120$

Solve for t = the total number of families.

The coefficient of t is $40\%\text{or}\frac{40}{100}$

The reciprocal of the coefficient of t is $\frac{40}{100}$ and $\frac{100}{40}·\frac{40}{100}=1$

Step 1: We need to multiply each side by the reciprocal of the coefficient of t

  $\begin{array}{l}\frac{40}{100}·t=120\\ \frac{100}{40}·\frac{40}{100}·t=\frac{100}{40}·120[\text{Multiply}\text{both}\text{sides}\text{by}\text{the}\text{reciprocal}\text{of}\text{the}\text{coefficient}\text{of}t]\end{array}$

Step 2: Simplify $t=\frac{12000}{40}\Rightarrow t=300$

Thus, multiply each side by the reciprocal of the coefficient of t

(f)

To determine

Find number of more families stayed at the campsite in July than in August.

Answer to Problem 83PFA

  $20$ more families stayed in the campsite

Explanation of Solution

Given:

At a camp site in July, 120 families stayed for three or more days. The 120 families represent three − eighths of all the families who stayed at the campsite in July. In August, 120 families stayed for three more days and represent 40% of all the families who stayed at the campsite in August.

Concept Used:

In July: Equation: $\frac{3}{8}·t=120$

In August: Equation: $40\%·t=120$

Calculation:

In July: Equation: $\frac{3}{8}·t=120$

Solve fort = total number of family member.

  $\begin{array}{l}\frac{3}{8}·t=120\\ \frac{8}{3}·\frac{3}{8}·t=\frac{8}{3}·120\text{[}\text{Multiply}\text{both}\text{sides}\text{by}\text{the}\text{reciprocal}\text{of}\text{the}\text{coefficent}\text{of}t\text{]}\\ \text{t}\text{=}\frac{960}{3}[\frac{8}{3}·\frac{3}{8}=1]\\ t=320\end{array}$

Total number of family member in July is 320

In August: Equation: $40\%·t=120$

Solve for t = the total number of families.

The coefficient of t is $40\%\text{or}\frac{40}{100}$

The reciprocal of the coefficient of t is $\frac{40}{100}$ and $\frac{100}{40}·\frac{40}{100}=1$

  $\begin{array}{l}\frac{40}{100}·t=120\\ \frac{100}{40}·\frac{40}{100}·t=\frac{100}{40}·120[\text{Multiply}\text{both}\text{sides}\text{by}\text{the}\text{reciprocal}\text{of}\text{the}\text{coefficient}\text{of}t]\\ t=\frac{12000}{40}[\frac{100}{40}·\frac{40}{100}=1]\\ t=300\end{array}$

The total number of families stayed in August = 300

The total number of families stayed in July = 320

The total number of families stayed in August = 300

In July $320-300=20$more families stayed in the campsite.

Thus, in July $20$ more families stayed in the campsite