Ratio problems solved with an equation

Ratio, proportion and percent, grade 8 Lesson 4 of 12Learning

Ratio problems solved with an equation

Quantities in a ratio:

\displaystyle 3k

and

\displaystyle 5k

In the ratio $\displaystyle 3:5$ one quantity is 3 equal parts, and the other 5 such parts.
Let $\displaystyle k$ be the size of one part.

The first quantity: $\displaystyle 3k$ .
The second quantity: $\displaystyle 5k$ .
Their ratio is $\displaystyle 3:5$ whatever $\displaystyle k$ is: for example, for $\displaystyle {k}={4}$ the quantities are 12 and 20, and $\displaystyle {12:20}={3:5}$ .

A ratio problem with the sum given

A year group has 96 students.
The ratio of boys to girls is $\displaystyle 3:5$ .
How many boys and how many girls are in the year group?

There are $\displaystyle 3k$ boys and $\displaystyle 5k$ girls, and together they are 96:

3k+5k=96

$\displaystyle {8k}={96}$ , so $\displaystyle {k}={12}$ .
Boys: $\displaystyle {3\cdot12}={36}$ ; girls: $\displaystyle {5\cdot12}={60}$ .
Check: $\displaystyle {36+60}={96}$ .

Exercise 1

The ratio of two numbers is $\displaystyle 4:5$ .
Their sum is 72.
Find the two numbers.

First 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

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

A ratio problem with the difference given

The ratio of Danny’s money to Maya’s is $\displaystyle 2:7$ .
Maya has 25 ₪ more than Danny.
How much money does each have?

Danny has $\displaystyle 2k$ shekels, and Maya $\displaystyle 7k$ . The difference between them is 25:

7k-2k=25

$\displaystyle {5k}={25}$ , so $\displaystyle {k}={5}$ .
Danny has $\displaystyle {2\cdot5}={10}$ ₪, and Maya $\displaystyle {7\cdot5}={35}$ ₪.
Check: $\displaystyle {35-10}={25}$ .

Exercise 2

In a library the ratio of reading books to textbooks is $\displaystyle 9:4$ .
It has 150 more reading books than textbooks.
How many books of each kind does the library have?

Reading books

/ – 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

Textbooks

/ – 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 ratio problem where the ratio changes

A box held red and blue balls in the ratio $\displaystyle 4:3$ .
6 red balls were taken out, and now the box has as many red balls as blue ones.
How many balls were in the box at first?

At first there were $\displaystyle 4k$ red and $\displaystyle 3k$ blue. After 6 red were taken out, $\displaystyle 4k-6$ red were left, as many as the blue:

4k-6=3k

$\displaystyle {k}={6}$ .
At first: $\displaystyle {4\cdot6}={24}$ red and $\displaystyle {3\cdot6}={18}$ blue – 42 balls together.
Check: after taking them out, $\displaystyle {24-6}={18}$ red, as many as the blue.

Exercise 3

The angles of a triangle are in the ratio $\displaystyle 2:3:4$ .
Find the angles.

Smallest 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

Middle 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

Largest 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 a class the ratio of boys to girls was $\displaystyle 5:4$ .
3 boys moved to another class, and now the class has as many boys as girls.
How many students were in the class 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

Formulas: ratio, proportion and percent, grade 8

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Ratio and proportion

A ratio is also a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$
Quantities in the ratio $\displaystyle a:b$ – $\displaystyle k$ the size of one part
$\displaystyle ak:bk$ $\displaystyle =$ $\displaystyle a:b$
Directly proportional quantities – the ratio $\displaystyle k$ stays the same (the unit price, for one)
$\displaystyle \frac{y}{x}=k$

Scale

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\cdot n$
An area at the scale $\displaystyle 1:n$ – it grows $\displaystyle n^2$ times, not $\displaystyle n$ times
$\displaystyle S_b$ $\displaystyle =$ $\displaystyle S_a\cdot n^2$

Percent and percent change

$\displaystyle p$ percent are $\displaystyle p$ hundredths
$\displaystyle p\%$ $\displaystyle =$ $\displaystyle \frac{p}{100}$
A rise of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot a$
A fall of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
The percent change – a part of the amount before the change
$\displaystyle p\%=\frac{b-a}{a}$
Two changes one after the other – a rise of $\displaystyle p\%$ , then a rise of $\displaystyle q\%$ (for a fall: minus instead of plus)
$\displaystyle c$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot(100\%+q\%)\cdot a$