Linear Equations Made Great : With Word Problems For Class 7th,8th,9th,10th[5]

Q.1) Amit travelled 2 / 5 of his journey by train, 1 / 3 by taxi, 1 / 6 by bus and remaining 6 km on foot. What is the length of his journey ?

SOLUTION:

Given:
Let the length of his journey = x km
Journey by train = ( 2x / 5 ) km
Journey by taxi = ( 1x / 3 ) km
Journey by bus = ( 1x / 6 ) km
Journey on foot = 6 km

According to the problem,

2x / 5 + 1x / 6 + 1x / 3 + 6 / 1 = x / 1
12x / 30 + 5x / 30 + 10x / 30 + 180 / 30 = 30x / 30
27x + 180 = 30x
30x – 27x = 180
3x = 180
x = 180 / 3
x = 60

Thus,
The length of journey = x = 60 km
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Q.2) Fifteen years later, Shreya will be four times as old as she was fifteen years ago. How old is she now ?

SOLUTION:

Given:
Let Shreya’s present age be = x years
Fifteen years ago, Shreya’s age = ( x – 15 ) years
Fifteen years later, Shreya’s age = ( x + 15 ) years

According to the problem,

( x + 15 ) = 4 ( x – 15 )
x + 15 = 4x – 60
15 + 60 = 4x – 1x
75 = 3x
75 / 3 = x
25 = x

( x = 25 )

Thus,
Shreya’s present age = 25 years
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Q.3) A number consists of 2 digits whose sum is 8. If 18 added to the number, the digits interchange their places. Find the number.

SOLUTION:

Given:
Let the digit in the Units place be = x
Let the digit in the Tens place be = ( 8 – x)

STEP – 1) i.e., Tens Units
( 8 – x) x

Hence, the original no. is 10 ( 8 – x ) + x
= 80 – 10x + x
= ( 80 – 9x )

STEP – 2) The interchanged digits are
In the Unit’s place = ( 8 – x )
In the Ten’s place = x

i.e., Tens Units
x ( 8 – x)

Hence, the interchanged no. = 10x + 8 – x
= 10x – x + 8
= ( 9x + 8 )

According to the problem,

18 + ( 80 – 9x ) = 9x + 8
18 + 80 – 9x = 9x + 8
98 – 9x = 9x + 8
98 – 8 = 9x + 9x
90 = 18x
90 / 18 = x
5 = x

( x = 5 )

Thus,
The original no. = 80 – 9x
= 80 – 9 * 5
= 80 – 45
= 35
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Q.4) Three numbers are in the ratio 4 : 5 : 7. If the sum of the largest and the smallest number is 72 more than the third number, find the numbers.

SOLUTION:

Given:
Let the first no. be = 4x
Let the second no. be = 5x
Let the third no. be = 7x

According to the problem,

( 7x + 4x ) = ( 5x + 72 )
11x = 5x + 72
11x – 5x = 72
6x = 72
x = 72 / 6
x = 12

Thus,
The first no. = 4x = 4 * 12 = 48
The second no. = 5x = 5 * 12 = 60
The third no. = 7x = 7 * 12 = 84
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Q.5) Two cars start from the same place but in opposite directions. Their speeds are 50 km/h and 65 km/h, respectively. After how long will they be 345 km apart ?

SOLUTION:

Given:
First Car Second Car
Speed = 50 km/h Speed = 65 km/h

The time taken for the cars to be 345 km apart = t

For first car For Second Car
Speed = d₁ / t Speed = d₂ / t
d₁ = speed * t d₂ = speed * t
d₁ = 50t d₂ = 65t

But we know,

d₁ + d₂ = 345 km
50t + 65t = 345
115t = 345
t = 345 / 115
t = 3 hours

Thus,
The time taken by both the cars to be 345 km apart is 3 hours.
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