Percent problems by an equation

Algebra, grade 8 Lesson 16 of 27Learning

Percent problems by an equation

Percent of

\displaystyle x

20% of $\displaystyle x$ is $\displaystyle 0.2x$ , and $\displaystyle p$ percent of $\displaystyle x$ is $\displaystyle \frac{p}{100}\cdot x$ . So a quantity a percent more or less than $\displaystyle x$ is an expression in $\displaystyle x$ too:

$\displaystyle x$ plus 20% of it: $\displaystyle {x+0.2x}={1.2x}$ .
$\displaystyle x$ minus 20% of it: $\displaystyle {x-0.2x}={0.8x}$ .
25% more than $\displaystyle x$ : $\displaystyle 1.25x$ .
40% less than $\displaystyle x$ : $\displaystyle 0.6x$ .

Exercise 1

A number plus 15% of it is 230.
What is the number?

Example: sharing in percent

Dan and Ron got 450 ₪ together for work in a garden.
Ron got 25% more than Dan.
How much did each of them get?

$\displaystyle x$ – Dan’s amount, in shekels. Ron got $\displaystyle 1.25x$ , and together they got 450:

\begin{aligned}x+1.25x&=450\\ 2.25x&=450 &&/:2.25\\ x&=200\end{aligned}

Answer: Dan got 200 ₪, and Ron $\displaystyle {1.25\cdot200}={250}$ ₪. Check: $\displaystyle {200+250}={450}$ .

Exercise 2

Maya and Tal shared 360 stickers.
Tal got 20% fewer than Maya.
How many stickers did each of them get?

Maya

/ – 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

Tal

/ – 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 discount plus a fixed amount

The discount is on the shoes only – not on the delivery.

Noa ordered shoes online at a 20% discount.
She paid another 30 ₪ for delivery.
Altogether she paid 270 ₪.
What was the price of the shoes before the discount?

$\displaystyle x$ – the price before the discount. After the discount the shoes cost $\displaystyle 0.8x$ , and with the delivery $\displaystyle 0.8x+30$ :

\begin{aligned}0.8x+30&=270 &&/-30\\ 0.8x&=240 &&/:0.8\\ x&=300\end{aligned}

Answer: 300 ₪. Check: $\displaystyle {0.8\cdot300+30}={240+30}={270}$ .

Exercise 3

Yoav bought a coat at a 25% discount, and a scarf at full price – 40 ₪.
Altogether he paid 190 ₪.
How much did the coat cost before the discount?

/ – 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 salary raise

Rotem’s salary went up by 10%, and on top of it she gets another 200 ₪ a month for travel.
Now she gets 7900 ₪ a month.
What was her salary before the raise?

$\displaystyle x$ – the salary before the raise. After the raise the salary is $\displaystyle 1.1x$ :

\begin{aligned}1.1x+200&=7900 &&/-200\\ 1.1x&=7700 &&/:1.1\\ x&=7000\end{aligned}

Answer: 7000 ₪. Check: $\displaystyle {1.1\cdot7000}={7700}$ , and 200 more is 7900.

Exercise 4

Yael spent 30% of her money on a book, and another 45 ₪ on a cinema ticket.
She had 25 ₪ left.
How much money did she have at first?

/ – 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: two prices that became equal

\displaystyle 0.8(x+60)

: the discount is on the whole price of shirt A.

Shirt A cost 60 ₪ more than shirt B.
Shirt A went down by 20%, and shirt B went up by 10%.
Now the two shirts cost the same.
What did each shirt cost at first?

$\displaystyle x$ – the price of shirt B at first; $\displaystyle x+60$ – the price of shirt A. The new prices are equal:

\begin{aligned}0.8(x+60)&=1.1x\\ 0.8x+48&=1.1x &&/-0.8x\\ 48&=0.3x &&/:0.3\\ 160&=x\end{aligned}

Answer: shirt B cost 160 ₪, and shirt A 220 ₪. Check: $\displaystyle {0.8\cdot220}={176}$ , and $\displaystyle {1.1\cdot160}={176}$ .

Exercise 5

In shop A, headphones cost 100 ₪ more than in shop B.
Shop A lowered their price by 20%, and shop B raised it by 20%.
Now the price is the same in both shops.
What was the price in each shop at first?

Shop A

/ – 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

Shop B

/ – 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

60% of the students in a class are girls.
There are 6 more girls than boys in the class.
How many students are in the class?

/ – 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 product’s price with VAT (18%) is 36 ₪ more than its price without VAT.
What is the product’s price with VAT?

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