Word problems

Algebra, grade 9 Lesson 25 of 26Learning

Word problems

A word problem becomes an equation, and many problems lead to a quadratic equation or to a system. The quadratic equation often has two solutions, and the story decides which of them is the answer.

The steps of solving a word problem

A solution that does not fit the story – a negative length, half a person – is rejected.

Choose the variable, and write what it stands for: “ $\displaystyle x$ – the width in cm”.
Write the other amounts with $\displaystyle x$ .
Build the equation from the sentence that says what is equal.
Solve.
Check each solution against the story, and answer the question in words.

Example: numbers

The product of two consecutive natural numbers is 56.
What are they?

$\displaystyle x$ – the smaller number, $\displaystyle x+1$ – the next one. $\displaystyle {x(x+1)}={56}$ , $\displaystyle {x^{2}+x-56}={0}$ , $\displaystyle {(x+8)(x-7)}={0}$ , so $\displaystyle {x}={7}$ or $\displaystyle {x}={-8}$ . $\displaystyle -8$ is not a natural number and is rejected. The numbers are 7 and 8 ( $\displaystyle {7\cdot8}={56}$ ).

Example: the area of a rectangle

A rectangle is 3 cm longer than it is wide, and its area is 40 cm².
What are its sides?

$\displaystyle x$ – the width, $\displaystyle x+3$ – the length. $\displaystyle {x(x+3)}={40}$ , $\displaystyle {x^{2}+3x-40}={0}$ , $\displaystyle {(x+8)(x-5)}={0}$ , so $\displaystyle {x}={5}$ or $\displaystyle {x}={-8}$ . A length is positive, so $\displaystyle -8$ is rejected. The sides: 5 cm and 8 cm.

Example: motion

distance = speed × time, so time = distance : speed.

A cyclist rode 60 km.
Had he ridden 5 km/h faster, he would have arrived an hour earlier.
What was his speed?

	speed (km/h)	time (h)	distance (km)
as he rode	$\displaystyle v$	$\displaystyle \frac{60}{v}$	60
faster	$\displaystyle v+5$	$\displaystyle \frac{60}{v+5}$	60

The faster ride takes one hour less: $\displaystyle {\frac{60}{v}-\frac{60}{v+5}}={1}$ . Multiplying by $\displaystyle v(v+5)$ : $\displaystyle {60(v+5)-60v}={v(v+5)}$ , that is $\displaystyle {v^{2}+5v-300}={0}$ . The formula: $\displaystyle {v}={\frac{-5\pm\sqrt{25+1200}}{2}}={\frac{-5\pm35}{2}}$ , so $\displaystyle {v}={15}$ or $\displaystyle {v}={-20}$ . A speed is positive: the cyclist rode at 15 km/h. Check: 60 km take 4 hours at 15 km/h and 3 hours at 20 km/h.

Two unknowns: a system

When the story has two unknown amounts, each can have its own variable, and each sentence gives an equation. A rectangle’s perimeter is 34 cm and its diagonal is 13 cm: with sides $\displaystyle x$ and $\displaystyle y$ , $\displaystyle {x+y}={17}$ and (by the Pythagorean theorem) $\displaystyle {x^{2}+y^{2}}={169}$ . Isolate $\displaystyle {y}={17-x}$ and substitute it into the second equation.

Exercise 1

The square of a natural number is 30 more than the number.
What is the number?

The number

/ – 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

Exercise 2

In a right-angled triangle one leg is 7 cm longer than the other, and the hypotenuse is 13 cm.
Find the legs.

The shorter leg

/ – 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 longer leg

/ – 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 3

A picture 20 cm by 30 cm has a frame of the same width all round it.
The area of the picture with the frame is 1200 cm².
How wide is the frame?
Draw it first.

The frame’s 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

Exercise 4

Find the sides of the rectangle whose perimeter is 34 cm and diagonal 13 cm (the example above).

Exercise 5

The sum of two numbers is 10 and their product is 21.
Write a system of equations and solve it.

Exercise 6

A car drove 240 km from Haifa to Eilat.
On the way back it drove 20 km/h faster, and the trip took one hour less.
What was its speed on the way there?
Make a table like the one in the example.

The speed on the way there

/ – 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 7

A ball is thrown straight up.
Its height in metres after $\displaystyle t$ seconds is $\displaystyle {h}={-5t^{2}+20t}$ .
a. When is the ball 15 m high? Why are there two answers?
b. When does it land?
c. What is the greatest height it reaches, and when?

b. The landing

/ – 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

c. The greatest height

/ – 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

c. When

/ – 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 coat cost 200 shekels.
Its price went up twice by the same percentage, and now it costs 242 shekels.
By what percentage did it go up each time?
Hint: a rise of $\displaystyle p$ percent multiplies the price by $\displaystyle 1+\frac{p}{100}$ .

The percent

/ – 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 9

Open as a page

The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$