APPLYING RATIOS AND RATES

About "Applying ratios and rates"

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.  

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 "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

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

Apart from the stuff given above, if you want to know more about "Applying ratios and rates", please click here

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

HTML Comment Box is loading comments...

ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula

Solving quadratic equations by completing square

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

Algebraic identities

Solving absolute value equations 

Solving Absolute value inequalities

Graphing absolute value equations  

Combining like terms

Square root of polynomials 

HCF and LCM 

Remainder theorem

Synthetic division

Logarithmic problems

Simplifying radical expression

Comparing surds

Simplifying logarithmic expressions

Negative exponents rules

Scientific notations

Exponents and power

COMPETITIVE EXAMS

Quantitative aptitude

Multiplication tricks

APTITUDE TESTS ONLINE

Aptitude test online

ACT MATH ONLINE TEST

Test - I

Test - II

TRANSFORMATIONS OF FUNCTIONS

Horizontal translation

Vertical translation

Reflection through x -axis

Reflection through y -axis

Horizontal expansion and compression

Vertical  expansion and compression

Rotation transformation

Geometry transformation

Translation transformation

Dilation transformation matrix

Transformations using matrices

ORDER OF OPERATIONS

BODMAS Rule

PEMDAS Rule

WORKSHEETS

Converting customary units worksheet

Converting metric units worksheet

Decimal representation worksheets

Double facts worksheets

Missing addend worksheets

Mensuration worksheets

Geometry worksheets

Comparing  rates worksheet

Customary units worksheet

Metric units worksheet

Complementary and supplementary worksheet

Complementary and supplementary word problems worksheet

Area and perimeter worksheets

Sum of the angles in a triangle is 180 degree worksheet

Types of angles worksheet

Properties of parallelogram worksheet

Proving triangle congruence worksheet

Special line segments in triangles worksheet

Proving trigonometric identities worksheet

Properties of triangle worksheet

Estimating percent worksheets

Quadratic equations word problems worksheet

Integers and absolute value worksheets

Decimal place value worksheets

Distributive property of multiplication worksheet - I

Distributive property of multiplication worksheet - II

Writing and evaluating expressions worksheet

Nature of the roots of a quadratic equation worksheets

Determine if the relationship is proportional worksheet

TRIGONOMETRY

SOHCAHTOA

Trigonometric ratio table

Problems on trigonometric ratios

Trigonometric ratios of some specific angles

ASTC formula

All silver tea cups

All students take calculus 

All sin tan cos rule

Trigonometric ratios of some negative angles

Trigonometric ratios of 90 degree minus theta

Trigonometric ratios of 90 degree plus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 180 degree minus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 270 degree minus theta

Trigonometric ratios of 270 degree plus theta

Trigonometric ratios of angles greater than or equal to 360 degree

Trigonometric ratios of complementary angles

Trigonometric ratios of supplementary angles 

Trigonometric identities 

Problems on trigonometric identities 

Trigonometry heights and distances

Domain and range of trigonometric functions 

Domain and range of inverse  trigonometric functions

Solving word problems in trigonometry

Pythagorean theorem

MENSURATION

Mensuration formulas

Area and perimeter

Volume

GEOMETRY

Types of angles 

Types of triangles

Properties of triangle

Sum of the angle in a triangle is 180 degree

Properties of parallelogram

Construction of triangles - I 

Construction of triangles - II

Construction of triangles - III

Construction of angles - I 

Construction of angles - II

Construction angle bisector

Construction of perpendicular

Construction of perpendicular bisector

Geometry dictionary

Geometry questions 

Angle bisector theorem

Basic proportionality theorem

ANALYTICAL GEOMETRY

Analytical geometry formulas

Distance between two points

Different forms equations of straight lines

Point of intersection

Slope of the line 

Perpendicular distance

Midpoint

Area of triangle

Area of quadrilateral

Parabola

CALCULATORS

Matrix Calculators

Analytical geometry calculators

Statistics calculators

Mensuration calculators

Algebra calculators

Chemistry periodic calculator

MATH FOR KIDS

Missing addend 

Double facts 

Doubles word problems

LIFE MATHEMATICS

Direct proportion and inverse proportion

Constant of proportionality 

Unitary method direct variation

Unitary method inverse variation

Unitary method time and work

SYMMETRY

Order of rotational symmetry

Order of rotational symmetry of a circle

Order of rotational symmetry of a square

Lines of symmetry

CONVERSIONS

Converting metric units

Converting customary units

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic 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