Word problems: numbers, ages and money

Algebra, grade 8 Lesson 15 of 27Learning

Word problems: numbers, ages and money

The five steps

Write what

\displaystyle x

stands for, with its units: “

\displaystyle x

– Omer’s age today, in years”.

A word problem is solved in five steps:

Choose the unknown: what $\displaystyle x$ stands for.
Express the other quantities with $\displaystyle x$ .
Write an equation from a sentence in the story that says two quantities are equal.
Solve the equation.
Answer the question and check in the story. Sometimes the question is about another quantity, not about $\displaystyle x$ .

From words to expressions

Careful: “5 more” is adding or subtracting, and “3 times” – multiplying or dividing.

The same words come back in many stories. When one quantity is $\displaystyle x$ :

In the story	The expression
5 more than $\displaystyle x$	$\displaystyle x+5$
5 less than $\displaystyle x$	$\displaystyle x-5$
3 times $\displaystyle x$	$\displaystyle 3x$
a third of $\displaystyle x$	$\displaystyle \frac{x}{3}$
two numbers whose sum is 40; one of them is $\displaystyle x$	the other: $\displaystyle 40-x$
the age in 4 years	$\displaystyle x+4$
the age 4 years ago	$\displaystyle x-4$

Exercise 1

Danny has $\displaystyle x$ shekels.
Maya has 3 times as much as Danny.
Yoni has 20 ₪ less than Maya.
Write an expression for Maya’s money and one for Yoni’s money.
Write an expression for the money of all three together, and simplify it.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: numbers

One number is 3 more than another.
4 times the smaller number equals 2 times the larger number plus 10.
What are the numbers?

$\displaystyle x$ – the smaller number; $\displaystyle x+3$ – the larger number. The second sentence is the equation:

\begin{aligned}4x&=2(x+3)+10\\ 4x&=2x+16 &&/-2x\\ 2x&=16 &&/:2\\ x&=8\end{aligned}

Answer: the numbers are 8 and 11. Check: $\displaystyle {4\cdot8}={32}$ , and $\displaystyle {2\cdot11+10}={32}$ .

Exercise 2

The sum of two numbers is 40.
3 times the first number is 8 more than the second number.
What are the numbers?

The first number

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The second number

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: ages now and later

In 12 years each of them is 12 years older.

Itai’s mother is 3 times as old as he is.
In 12 years she will be 2 times as old as he is.
How old are they today?

$\displaystyle x$ – Itai’s age today. A table puts the ages in order:

	Today	In 12 years
Itai	$\displaystyle x$	$\displaystyle x+12$
Mother	$\displaystyle 3x$	$\displaystyle 3x+12$

In 12 years, the mother’s age equals 2 times Itai’s age:

\begin{aligned}3x+12&=2(x+12)\\ 3x+12&=2x+24 &&/-2x\\ x+12&=24 &&/-12\\ x&=12\end{aligned}

Answer: Itai is 12, and his mother is 36. Check: in 12 years they will be 24 and 48, and 48 is 2 times 24.

Exercise 3

Omer’s father is 4 times as old as Omer.
5 years ago the father was 7 times as old as Omer.
How old are they today?
Make a table as in the example, write an equation and solve it.

Omer

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

His father

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: savings

Alon has 120 ₪, and he saves 15 ₪ every week.
Gal has 200 ₪, and she spends 5 ₪ every week.
After how many weeks will they have the same amount?

$\displaystyle x$ – the number of weeks. After $\displaystyle x$ weeks Alon will have $\displaystyle 120+15x$ ₪, and Gal $\displaystyle 200-5x$ ₪:

\begin{aligned}120+15x&=200-5x &&/+5x\\ 120+20x&=200 &&/-120\\ 20x&=80 &&/:20\\ x&=4\end{aligned}

Answer: after 4 weeks, and then each of them will have 180 ₪. Check: $\displaystyle {120+15\cdot4}={180}$ , and $\displaystyle {200-5\cdot4}={180}$ .

Exercise 4

Three friends bought a present for 140 ₪.
Shira paid 2 times as much as Tom.
Ido paid 10 ₪ less than Shira.
How much did each of them pay?

Tom

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Shira

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Ido

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Maya has 300 ₪, and she spends 20 ₪ every week.
Omri has 60 ₪, and he saves 40 ₪ every week.
After how many weeks will they have the same amount?
How much money will each of them have then?

The number of weeks

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Each of them then (₪)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 6

Dana thought of a number.
When she multiplies it by 3 and adds 8, she gets the same result as when she subtracts 4 from it and multiplies by 5.
What number did Dana think of?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: algebra, grade 8

Open as a page

The linear function

A linear function – $\displaystyle m$ the slope; the line crosses the $\displaystyle y$ -axis at $\displaystyle (0,b)$
$\displaystyle y=mx+b$
The slope of the line through $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Parallel lines have the same slope
$\displaystyle m_1=m_2$
A line through the origin – $\displaystyle {b}={0}$
$\displaystyle y=mx$
A horizontal line – the slope is 0
$\displaystyle y=b$

Word problems

$\displaystyle p$ percent of $\displaystyle x$
$\displaystyle \frac{p}{100}\cdot x$
$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$

Inequalities

Adding or subtracting the same number on both sides – the sign stays
$\displaystyle a+c<b+c$ $\displaystyle a<b$
Multiplying or dividing by a positive number – the sign stays
$\displaystyle a\cdot c<b\cdot c$ $\displaystyle a<b,\ c>0$
Multiplying or dividing by a negative number – the sign turns round
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$