Word problems

Algebra, grade 7 Lesson 20 of 22Learning

Word problems

Solving a word problem

Check in the story – not only in the equation.

In a word problem you look for an amount you do not know. Find it in five steps:

Choose what to call $\displaystyle x$ – usually the amount the question asks about, or the smallest of the amounts.
Express the other amounts using $\displaystyle x$ .
Write an equation from what is known about the amounts.
Solve the equation.
Check the answer in the story, and answer in a full sentence.

Example: ages

Dana is 4 years older than her brother, and together they are 22.
How old are they?
Let $\displaystyle x$ be the brother’s age. Dana’s age is $\displaystyle x+4$ , and the ages add up to 22:

\begin{aligned}x+x+4&=22\\ 2x+4&=22 &&/-4\\ 2x&=18 &&/:2\\ x&=9\end{aligned}

Check in the story: the brother is 9, Dana is 13 – 4 years older than him, and together $\displaystyle {9+13}={22}$ . Answer: the brother is 9 and Dana is 13.

Exercise 1

The sum of three consecutive even numbers (even numbers that come one after another, like 4, 6, 8) is 66.
What are the numbers?
Let $\displaystyle x$ be the smallest of them – then the next one is $\displaystyle x+2$ .
Simplify the sum of the three numbers, then write the equation, solve and check.

/ – 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 smallest

/ – 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 middle one

/ – 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 largest

/ – 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

A notebook costs 3 ₪ more than a pencil.
Noa bought 2 notebooks and 5 pencils and paid 34 ₪.
How much does a pencil cost, and how much does a notebook cost?
Let $\displaystyle x$ be the price of a pencil, and simplify the price of the whole purchase.

/ – 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

A pencil

/ – 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

A notebook

/ – 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: perimeter

A rectangle is 3 cm longer than it is wide, and its perimeter is 30 cm.
Let $\displaystyle x$ be the width; the length is $\displaystyle x+3$ , and the perimeter is twice the width plus twice the length:

\begin{aligned}2x+2(x+3)&=30\\ 4x+6&=30 &&/-6\\ 4x&=24 &&/:4\\ x&=6\end{aligned}

Answer: the rectangle is 6 cm wide and 9 cm long. (Check: $\displaystyle {6+9+6+9}={30}$ .)

Exercise 3

An isosceles triangle has two equal sides, called legs; the third side is the base.
In this triangle the base is 4 cm shorter than each of the legs, and the perimeter of the triangle is 35 cm.
How long are the sides?
Simplify the perimeter.

/ – 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

A 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 base

/ – 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: motion

\displaystyle s

\displaystyle =

\displaystyle v\cdot t

$\displaystyle s$ – the distance, $\displaystyle v$ – the speed, $\displaystyle t$ – the time.

A car drove for some hours at 60 km/h, and then for another 3 hours at 40 km/h.
Altogether it covered 240 km.
How many hours did it drive at 60 km/h?
Call them $\displaystyle x$ . In the first part it covered $\displaystyle 60x$ km, and in the second $\displaystyle {40\cdot3}={120}$ km:

\begin{aligned}60x+120&=240 &&/-120\\ 60x&=120 &&/:60\\ x&=2\end{aligned}

Answer: two hours. (Check: $\displaystyle {120+120}={240}$ .)

Exercise 4

Two cyclists set out at the same time from two towns 90 km apart, riding towards each other.
One rides at 20 km/h and the other at 25 km/h.
After how many hours will they meet?
Hint: when they meet, together they have covered the whole distance.

/ – 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

Write a story that the equation $\displaystyle {3x+5}={20}$ fits.
What does $\displaystyle x$ stand for in your story?
Solve, and answer in a sentence.

Formulas: algebra, grade 7

Open as a page

Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$