Word problems: shapes, money and mixtures

Algebra, grade 8 Lesson 18 of 27Learning

Word problems: shapes, money and mixtures

Example: the perimeter of a triangle

In a triangle, one side is 2 times as long as the shortest side.
The third side is 5 cm longer than the shortest side.
The perimeter of the triangle is 37 cm.
How long are the sides?

$\displaystyle x$ – the shortest side, in cm; the other sides are $\displaystyle 2x$ and $\displaystyle x+5$ . The perimeter is the sum of the three sides:

\begin{aligned}x+2x+x+5&=37\\ 4x+5&=37 &&/-5\\ 4x&=32 &&/:4\\ x&=8\end{aligned}

Answer: the sides are 8 cm, 16 cm and 13 cm. Check: $\displaystyle {8+16+13}={37}$ .

Exercise 1

The length of a rectangle is 2 times its width plus 5 cm.
The perimeter of the rectangle is 46 cm.
What are the length and the width of the rectangle?

The length

/ – 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

The width

/ – 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: the area of rectangles

The area of a rectangle is its length times its width.

Two rectangles have the same width, 6 cm.
One rectangle is 4 cm longer than the other.
Together their area is 108 cm².
How long is each rectangle?

$\displaystyle x$ – the length of the shorter rectangle; the longer one’s length is $\displaystyle x+4$ . The areas are $\displaystyle 6x$ and $\displaystyle 6(x+4)$ :

\begin{aligned}6x+6(x+4)&=108\\ 12x+24&=108 &&/-24\\ 12x&=84 &&/:12\\ x&=7\end{aligned}

Answer: 7 cm and 11 cm. Check: $\displaystyle {6\cdot7+6\cdot11}={42+66}={108}$ .

Exercise 2

One rectangle is 8 cm wide, and a second rectangle is 6 cm wide.
The second rectangle is 3 cm longer than the first.
The two rectangles have the same area.
How long is each rectangle, and what is their area?
Sketch the two rectangles first.

The first rectangle’s length

/ – 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 rectangle’s length

/ – 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 area

/ – 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: the angles of a triangle

The angles of a triangle add up to

\displaystyle 180^{\circ}

In a triangle, one angle is 2 times the smallest angle.
The third angle is $\displaystyle 20^{\circ}$ more than the smallest angle.
What are the angles of the triangle?

$\displaystyle x$ – the smallest angle, in degrees; the others are $\displaystyle 2x$ and $\displaystyle x+20$ . The three angles add up to 180:

\begin{aligned}x+2x+x+20&=180\\ 4x+20&=180 &&/-20\\ 4x&=160 &&/:4\\ x&=40\end{aligned}

Answer: the angles are $\displaystyle 40^{\circ}$ , $\displaystyle 80^{\circ}$ and $\displaystyle 60^{\circ}$ . Check: $\displaystyle {40+80+60}={180}$ .

Exercise 3

In a triangle, the second angle is 3 times the first angle.
The third angle is $\displaystyle 30^{\circ}$ less than the second angle.
What are the angles of the triangle?

The first angle

/ – 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 angle

/ – 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 third angle

/ – 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 4

In an isosceles triangle the base angles are equal.
The apex angle is $\displaystyle 30^{\circ}$ more than each base angle.
What are the angles of the triangle?

A base angle

/ – 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 apex angle

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Two quantities with a known total:

\displaystyle x

and

\displaystyle 20-x

145 ₪ is the coins’ value, not their number.

When you know how many items there are together, one unknown is enough: if there are 20 coins, and $\displaystyle x$ of them are of one kind, then $\displaystyle 20-x$ of them are of the other kind.
A money box holds 20 coins of 5 ₪ and of 10 ₪.
Together it holds 145 ₪.
How many coins of each kind are in the money box?

$\displaystyle x$ – the number of 10 ₪ coins.

	Number of coins	Value of a coin (₪)	Total value (₪)
10 ₪	$\displaystyle x$	10	$\displaystyle 10x$
5 ₪	$\displaystyle 20-x$	5	$\displaystyle 5(20-x)$

\begin{aligned}10x+5(20-x)&=145\\ 5x+100&=145 &&/-100\\ 5x&=45 &&/:5\\ x&=9\end{aligned}

Answer: 9 coins of 10 ₪ and 11 coins of 5 ₪. Check: $\displaystyle {90+55}={145}$ .

Exercise 5

A purse holds 30 coins of 2 ₪ and of 5 ₪.
Together it holds 99 ₪.
How many coins of each kind are in the purse?

5 ₪ coins

/ – 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

2 ₪ coins

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Exercise 6

A regular cinema ticket costs 40 ₪, and a student ticket 25 ₪.
For one screening 120 tickets were sold, for 3600 ₪.
How many tickets of each kind were sold?

Full-price tickets

/ – 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

Student tickets

/ – 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: a mixture

The value of the mixture equals the sum of the values of what goes into it.

A shop mixes nuts at 60 ₪ a kg with raisins at 30 ₪ a kg.
It wants 10 kg of a mixture to be sold at 42 ₪ a kg.
How many kg of each kind are needed?

$\displaystyle x$ – the kg of nuts; of raisins $\displaystyle 10-x$ .

	kg	Price per kg (₪)	Value (₪)
Nuts	$\displaystyle x$	60	$\displaystyle 60x$
Raisins	$\displaystyle 10-x$	30	$\displaystyle 30(10-x)$
Mixture	10	42	$\displaystyle 420$

\begin{aligned}60x+30(10-x)&=420\\ 30x+300&=420 &&/-300\\ 30x&=120 &&/:30\\ x&=4\end{aligned}

Answer: 4 kg of nuts and 6 kg of raisins. Check: $\displaystyle {240+180}={420}$ .

Exercise 7

A café mixes coffee at 80 ₪ a kg with coffee at 50 ₪ a kg.
It wants 6 kg of a mixture at 60 ₪ a kg.
How many kg of each kind are needed?

Coffee at 80 ₪ (kg)

/ – 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

Coffee at 50 ₪ (kg)

/ – 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 8

A bottle holds 2 litres of a drink, and 30% of it is juice.
We want to add water so that 20% of the drink is juice.
How many litres of water need to be added?
Hint: adding water does not change the amount of juice.

/ – 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
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$