Mathematics » Equations and Inequalities » Word Problems Continued

# Problem Solving Strategy

## Problem Solving Strategy

2. What is the question and what do we need to solve for?

3. Assign variables to the unknown quantities, for example, $$x$$ and $$y$$.

4. Translate the words into algebraic expressions by rewriting the given information in terms of the variables.

5. Set up a system of equations.

6. Solve for the variables using substitution.

7. Check the solution.

## Optional Investigation: Simple word problems

Write an equation that describes the following real-world situations mathematically:

1. Mohato and Lindiwe both have colds. Mohato sneezes twice for each sneeze of Lindiwe’s. If Lindiwe sneezes $$x$$ times, write an equation describing how many times they both sneezed.

2. The difference of two numbers is $$\text{10}$$ and the sum of their squares is $$\text{50}$$. Find the two numbers.

3. Liboko builds a rectangular storeroom. If the diagonal of the room is $$\sqrt{ \text{1 312}}$$ $$\text{m}$$ and the perimeter is $$\text{80}$$ $$\text{m}$$, determine the dimensions of the room.

4. It rains half as much in July as it does in December. If it rains $$y$$ mm in July, write an expression relating the rainfall in July and December.

5. Zane can paint a room in $$\text{4}$$ hours. Tlali can paint a room in $$\text{2}$$ hours. How long will it take both of them to paint a room together?

6. $$\text{25}$$ years ago, Arthur was $$\text{5}$$ years more than a third of Bongani’s age. Today, Bongani is $$\text{26}$$ years less than twice Arthur’s age. How old is Bongani?

7. The product of two integers is $$\text{95}$$. Find the integers if their total is $$\text{24}$$.

## Example

### Question

The annual gym subscription for a single member is $$\text{R}\,\text{1 000}$$, while an annual family membership is $$\text{R}\,\text{1 500}$$. The gym is considering increasing all membership fees by the same amount. If this is done then a single membership would cost $$\dfrac{5}{7}$$ of a family membership. Determine the amount of the proposed increase.

### Identify the unknown quantity and assign a variable

Let the amount of the proposed increase be $$x$$.

### Use the given information to complete a table

 now after increase single $$\text{1 000}$$ $$\text{1 000} + x$$ family $$\text{1 500}$$ $$\text{1 500} + x$$

### Set up an equation

$\text{1 000} + x = \cfrac{5}{7}(\text{1 500} + x)$

### Solve for $$x$$

\begin{align*} \text{7 000} + 7x &= \text{7 500} + 5x \\ 2x &= 500 \\ x &= 250 \end{align*}

The proposed increase is $$\text{R}\,\text{250}$$.

## Example

### Question

Erica has decided to treat her friends to coffee at the Corner Coffee House. Erica paid $$\text{R}\,\text{54.00}$$ for four cups of cappuccino and three cups of filter coffee. If a cup of cappuccino costs $$\text{R}\,\text{3.00}$$ more than a cup of filter coffee, calculate how much a cup of each type of coffee costs?

### Method 1: identify the unknown quantities and assign two variables

Let the cost of a cappuccino be $$x$$ and the cost of a filter coffee be $$y$$.

### Use the given information to set up a system of equations

\begin{align*} 4x + 3y &= 54 \qquad \ldots (1) \\ x &= y + 3 \qquad \ldots (2) \end{align*}

### Solve the equations by substituting the second equation into the first equation

\begin{align*} 4(y+3) + 3y &= 54 \\ 4y+12 + 3y &= 54 \\ 7y &= 42 \\ y &= 6 \end{align*}

If $$y=6$$, then using the second equation we have \begin{align*} x &= y + 3 \\ &= 6 + 3 \\ &= 9 \end{align*}

### Check that the solution satisfies both original equations

A cup of cappuccino costs $$\text{R}\,\text{9}$$ and a cup of filter coffee costs $$\text{R}\,\text{6}$$.

### Method 2: identify the unknown quantities and assign one variable

Let the cost of a cappuccino be $$x$$ and the cost of a filter coffee be $$x-3$$.

### Use the given information to set up an equation

$4x + 3(x-3) = 54$

### Solve for $$x$$

\begin{align*} 4x + 3(x-3) &= 54 \\ 4x + 3x-9 &= 54 \\ 7x &= 63 \\ x &= 9 \end{align*}

A cup of cappuccino costs $$\text{R}\,\text{9}$$ and a cup of filter coffee costs $$\text{R}\,\text{6}$$.

## Example

### Question

Two taps, one more powerful than the other, are used to fill a container. Working on its own, the less powerful tap takes $$\text{2}$$ hours longer than the other tap to fill the container. If both taps are opened, it takes $$\text{1}$$ hour, $$\text{52}$$ minutes and $$\text{30}$$ seconds to fill the container. Determine how long it takes the less powerful tap to fill the container on its own.

### Identify the unknown quantities and assign variables

Let the time taken for the less powerful tap to fill the container be $$x$$ and let the time taken for the more powerful tap be $$x – 2$$.

### Convert all units of time to be the same

First we must convert $$\text{1}$$ hour, $$\text{52}$$ minutes and $$\text{30}$$ seconds to hours: $1 + \cfrac{52}{60} + \cfrac{30}{(60)^2} = \text{1.875}\text{ hours}$

### Use the given information to set up a system of equations

Write an equation describing the two taps working together to fill the container: \begin{align*} \cfrac{1}{x} + \cfrac{1}{x-2} &= \cfrac{1}{\text{1.875}} \end{align*}

### Multiply the equation through by the lowest common denominator and simplify

\begin{align*} \text{1.875}(x-2) + \text{1.875}x &= x(x-2) \\ \text{1.875}x – \text{3.75} + \text{1.875}x &= x^2 – 2x \\ 0 &= x^2 – \text{5.75}x + \text{3.75} \end{align*}

Multiply the equation through by $$\text{4}$$ to make it easier to factorise (or use the quadratic formula) \begin{align*} 0 &= 4x^2 – 23x + 15 \\ 0 &= (4x-3)(x-5) \end{align*} Therefore $$x = \cfrac{3}{4}$$ or $$x = 5$$.

We have calculated that the less powerful tap takes $$\cfrac{3}{4}$$ hours or $$\text{5}$$ hours to fill the container, but we know that when both taps are opened it takes $$\text{1.875}$$ hours. We can therefore discard the first solution $$x = \dfrac{3}{4}$$ hours.

So the less powerful tap fills the container in $$\text{5}$$ hours and the more powerful tap takes $$\text{3}$$ hours.

### Check that the solution satisfies the original equation

The less powerful tap fills the container in $$\text{5}$$ hours and the more powerful tap takes $$\text{3}$$ hours.