TIME AND DISTANCE PROBLEMS

About "Time and Distance Problems"

Time and Distance Problems :

Problems on time and distance play a major role in competitive exams. It is bit difficult to score marks in competitive exams without knowing the stuff on time and distance. 

Before look at the problems, if you would like to know the shortcuts related to speed, distance and time, 

Please click here

Time and Distance Problems

Problem 1 : 

A person covers a certain distance at a certain speed. If he increases his speed by 33⅓ %, he takes 15 minutes less to cover the same distance. Find the time taken by him initially to cover the distance at the original speed.

Solution :

Let the original speed be 100%.

Given : The speed is increased by 33⅓ %.

Then, the speed after increment is 

133⅓ %

Ratio of the speeds is

100 %  :  133⅓ %

100  :  133

100  :  400/3

Divide both sides by 100. 

1  :  3/4

So, ratio of the speeds is

1 : 3/4

If the ratio of the speeds is 1 : 4/3, then the ratio of time taken to cover the same distance would be

1 : 3/4

When the speed is increased by 33⅓ %, 3/4 of the original time is enough to cover the same distance. 

That is, when the speed is increased by 33⅓ %, 1/4 of the original time will be decreased. 

The question says that when speed is increased by 33⅓ %, time is decreased by 15 minutes. 

So, we have

1/4 of the original time  =  15 minutes

Multiply both sides by 4.  

⋅ (1/4 of the original time)  =  (15 minutes) ⋅ 4

Original time  =  60 minutes

Original time  =  1 hour

Hence, the time taken by the person initially is 1 hour. 

Let us look at the next problem on "Time and distance problems".

Problem 2 :

A man traveled from the village to the post office at the rate of 25 k mph and walked back at the rate of 4 kmph. If the entire journey had taken 5 hours 48 minutes, find the distance of the post office from the village.

Solution :

Here, the distance covered in both the ways is same. 

So, the formula to find the average speed is

=  2pq / (p + q)

Plug p  =  25 and q  =  4.  

=  (2 ⋅ 25 ⋅ 4) / (25 + 4) 

=  200 / 29

The average speed is 200/29 km/hr.  

Given :The entire journey had taken 5 hours 48 minutes

5 hour 48 min  =  5 4860 hours

5 hours 48 min  =  5  hours

5 hours 48 min  =  29 / 5 hours

The formula to find the distance is

=  Speed ⋅ Time

Then, the distance covered in (29/5) hours at the average speed (200/29) kmph is

=  (200 / 29) ⋅ (29 / 5)

=  40 km 

So, the distance covered in the whole journey is 40 km.

(Whole journey : Village to post office + Post office to village)

Then the distance between the post office and village is 

=  40 / 2

=  20

Hence, the distance of the post office from the village is 20 km. 

Let us look at the next problem on "Time and distance problems".

Problem 3 :

If a man walks at the rate of 5 km/hr, he misses a train by 7 minutes. However, if he walks at the rate of 6 km/hr, he reaches the station 5 minutes before the arrival of the train. Find the distance covered by him to reach the station.

Solution :

Let "x" be the distance to be covered by the person to reach the station.  

The formula to find the time is

=  Distance / Speed

When the speed is 5 kmph, time is 

=  x/5  hrs

When the speed is 6 kmph, time is

=  x/6  hrs

Let "t" be the actual time required to cover the distance x.

And also,

7 minutes  =  7/60  hrs

5 minutes  =  5/60  =  1/12  hrs

Given : If the man walks at the rate of 5 km/hr, he misses the train by 7 minutes.

That is, he takes 7 minutes more than actual time. 

So, we have

t  =  x/5 - 7/60 ------(1)

Given : If he walks at the rate of 6 km/hr, he reaches the station 5 minutes before.

That is, he takes 5 minutes less than actual time. 

So, we have

t  =  x/6 + 1/12 ------(2)

From (1) and (2), we get

x/5  -  7/60  =  x/6  +  1/12

Solving for x :

12x/60  -  7/60  =  2x/12  +  1/12

(12x - 7) / 60  =  (2x + 1) / 12

L.C.M of (60, 12) is 60. 

Multiply both sides by 60.

12x - 7  =  5(2x + 1)

12x - 7  =  10x + 5

Simplify. 

2x  =  12

Divide both sides by 2. 

x  =  6

Hence,the distance covered by him to reach the station is 6 km.

Let us look at the next problem on "Time and distance problems".

Problem 4 : 

A person has to cover a distance of 6 miles in 45 minutes. If he covers one-half of the distance in two-thirds of the total time. What must his speed be to cover the remaining distance in the remaining time ?

Solution :

Given : Total distance is 6 miles and total time is 45 minutes. And also, he covers one-half of the distance in two-thirds of the total time.

One-half of the total distance 6 miles is 

=  ½ ⋅ 6

=  3 km

Two-thirds of the total time 45 minutes is 

=   ⋅ 45

=  30 minutes

From the above calculations, we have 

Remaining distance  =  6 - 3  =  3 miles

Remaining time  =  45 - 30  =  15 minutes

The formula to find the speed is 

=  Distance / Time

Plug, Distance  =  3 and Time  =  15.

=  3 / 15

=  1 / 5 miles per minute

=  (1/5) ⋅ 60 miles per hour

=  12 miles per hour. 

Hence, the speed must be 12 miles per hour. 

Let us look at the next problem on "Time and distance problems".

Problem 5 :

A is faster than B . A and B each walk 24 miles. The sum of their speeds is 7 miles per hour and the sum of their time taken is 14 hrs. Find A's speed and B's speed (in mph).

Solution :

Let "x" be the speed of A .

Then speed of B is

=  7 - x 

The formula to find time is

=  Distance / Speed

Then, time taken by A is 

=  24/x  hrs

Time taken by B is 

=  24/(7 - x)  hrs 

Given : Sum of time taken is 14 hours. 

So, we have 

24/x  +  24/( 7 - x)  =  14

L.C.M of x and (7 - x) is x(7 - x). 

Multiply both sides by x(7 - x). 

24 ⋅ (7-x) + 24 ⋅ x  =  14 ⋅ x(7 - x) 

168 - 24x + 24x  =  98x - 14x2

14x- 98x + 168  =  0

Divide both sides by 14.

x-  7x + 12  =  0

(x - 4)(x - 3)  =  0

x  =  4 or x  =  3

So, A's speed can be 4 mph or 3 mph. 

If A's speed is 4 mph, then B's speed

7 - x  =  7 - 4

7 - x  =  3 mph

If A's speed is 3 mph, then B's speed

7 - x  =  7 - 3

7 - x  =  4 mph

Given : A is faster than B

Hence, the speed of A is 4 miles per hour and B is 3 miles per hour. 

After having gone through the stuff given above, we hope that the students would have understood "Time and distance problems"

Apart from the stuff given above, if you want to know more about "Time and distance problems", please click here.

Apart from the stuff given in this section, 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