**Applying ratios and rates :**

In solving real world problems, we apply ratio, if we want to compare two quantities of same kind with same units.

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, we apply rates, if want to compare the given measure to one unit of another measure.

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

**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 "Applying 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 "Applying 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)

After the above new admissions,

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 "Applying 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 "Applying 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

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