# REPRESENTING RATIOS AND RATES

## About "Representing ratios and rates"

Representing ratios and rates :

Comparison of two quantities of same kind with same units is represented as a ratio.

For example, comparison of number of boys and girls in a school is represented as ratio between number of boys and girls.

On the other hand, comparison of the given measure to one unit of another measure is represented as rate.

For example, number of miles covered by a car in one hour is represented as rate.

## Applying ratios and rates - Examples

Example 1 :

The average age of three boys is 25 years and their ages are in the proportion 3:5:7. The age of the youngest boy is

Solution :

From the ratio 3 : 5 : 7, the ages of three boys are 3x, 5x and 7x.

Average age of three boys = 25

(3x+5x+7x)/3 = 25 ----------> 15x = 75 -----------> x = 5

Age of the first boy = 3x = 3(5) = 15

Age of the first boy = 5x = 5(5) = 25

Age of the first boy = 7x = 7(5) = 105

Hence the age of the youngest boy is 15 years.

Let us look at the next problem on "Representing ratios and rates"

Example 2 :

John weighs 56.7 kilograms. If he is going to reduce his weight in the ratio 7:6, find his new weight.

Solution :

Original weight of John = 56.7 kg (given)

He is going to reduce his weight in the ratio 7:6

His new weight = (6x56.7)/7 = 6x8.1 = 48.6 kg.

Hence his new weight = 48.6 kg

Let us look at the next problem on "Representing ratios and rates"

Example 3 :

The ratio of the no. of boys to the no. of girls in a school of 720 students is 3:5. If 18 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the no. of boys to the no. of girls may change to 2:3.

Solution :

Sum of the terms in the given ratio = 3+5 = 8

So, no. of boys in the school = 720x(3/8)= 270

No. of girls in the school = 720x(5/8)= 450

Let "x" be the no. of new boys admitted in the school.

No. of new girls admitted = 18  (given)

no. of boys in the school = 270+x

no. of girls in the school = 450+18 = 468

The ratio after the new admission is 2 : 3   (given)

So, (270+x) : 468  =  2 : 3

3(270+x)  =  468x2       (using cross product rule in proportion)

810 + 3x  =  936

3x  =  126

x  =  42

Hence the no. of new boys admitted in the school is 42

Let us look at the next problem on "Representing ratios and rates"

Example 4 :

The monthly incomes of two persons are in the ratio 4:5 and their monthly expenditures are in the ratio 7:9. If each saves \$50 per month, find the monthly income of the second person.

Solution :

From the given ratio of incomes ( 4 : 5 ),

Income of the 1st person = 4x

Income of the 2nd person = 5x

(Expenditure = Income - Savings)

Then, expenditure of the 1st person = 4x - 50

Expenditure of the 2nd person = 5x - 50

Expenditure ratio = 7 : 9  (given)

So, (4x - 50) : (5x - 50) = 7 : 9

9(4x - 50) = 7(5x - 50)

(using cross product rule in proportion)

36x - 450 = 35x - 350

x = 100

Then, income of the second person is

=  5x  =  5(100)  =  500.

Hence, income of the second person is \$500

Let us look at the next problem on "Representing ratios and rates"

Example 5 :

If the angles of a triangle are in the ratio 2:7:11, then find the angles.

Solution :

From the ratio 2 : 7 : 11,

the three angles are 2x, 7x, 11x

In any triangle, sum of the angles = 180

So, 2x + 7x + 11x  =  180°

20x  =  180 -------> x  =  9

Then, the first angle  =  2x  =  2(9)  = 18°

The second angle  =  7x  =  7(9)  =  63°

The third angle  =  11x  =  11(9)  99°

Hence the angles of the triangle are (18°, 63°, 99°)

Example 6 :

In a business, if A can earn \$ 7500 in 2.5 years, find the unit rate of his earning per month.

Solution :

Given : Earning in 2.5 years  =  \$ 7500

1 year  =  12 months

2.5 years  =  2.5 x 12  =  30 months

Then, earning in 30 months  =  \$ 7500

Therefore, earning in 1 month  =  7500 / 30  =  \$ 250

Hence, the unit rate of his earning per month is \$ 250

Example 7 :

If David can prepare 2 gallons of juice in 4 days, how many  cups of juice can he prepare per day ?

Solution :

No of gallons of juice prepared in 4 days  =  2 gallons

1 gallon  =  16 cups

So, no. of cups of juice prepared in 4 days  =  2 x 16  =  32 cups

Therefore, David can prepare 32 cups of juice in 4 days.

Then, no. of cups of juice prepared in 1 day  =  32 / 4  =  8

Hence, David can prepare 8 cups of juice in 1 day.

Example 8 :

If John can cover 360 miles in 3 hours, find the number of miles covered by John in 1 minute.

Solution :

No of miles covered in 3 hours  =  360

Then, no. of miles covered in 1 hour  =  360 / 3  =  180

1 hour  =  60 minutes

So, no. of miles covered in 60 minutes  =  180

Then, no. of miles covered 1 minute  =  180 / 60  =  3

Hence, John can cover 3 miles in 1 minute.

Example 9 :

Shanel walks 2/ 5 of a mile every 1/7 hour. Express her speed as a unit rate in miles per hour.

Solution :

Given : Shanel walks 2/ 5 of a mile every 1/7 hour

We know the formula for speed.

That is,  Speed  =  Distance / time

Speed  =  (2/5)  /  (1/7)

Speed  =  (2/5)  x  (7/1)

Speed  =  14 / 5

Speed  =  2.8 miles per hour.

Hence, the speed of Shanel is 2.8 miles per hour

Example 10 :

Declan use 2 /35 of a gallon of gas for every 4 /5 of a mile that he drives. At this rate, how many miles can he drive on one gallon of gas?

Solution :

Given : In 2 /35 of a gallon of gas, 4 /5 of a mile is traveled

Then, in  1 gallon of gas  =  (4/5) x (35/2) miles traveled.

=  14 miles traveled.

Hence, Declan can drive 14 miles in 1 gallon of gas

After having gone through the stuff given above, we hope that the students would have understood "Representing ratios and rates".

Apart from "Representing ratios and rates", if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6