## EXERCISE 15 B

Question 1

**The sum of three consecutive odd natural numbers is 87. What are the three numbers ?**

Sol :

Let the three consecutive odd numbers be (2x+1) , (2x+3) and (2x+5)

A.T.Q

⇒(2x+1)+(2x+3)+(2x+5)=87

⇒2x+1+2x+3+2x+5=87

⇒6x+9=87

⇒6x=87-9

⇒6x=78

⇒

⇒x=13

Hence , the consecutive numbers are

⇒2x+1=2×13+1=27 ,

⇒2x+3=2×13+3=29 and

⇒2x+5=2×13+5=31

Question 2

**The sum of two numbers is 80. If the larger number exceeds four times the smaller by 5 , what is the smaller number ?**

Sol :

Let the two numbers be x and y and lets assume x>y.

Also, x=4y+5 ..(i)

A.T.Q

x+y=80 ..(ii)

subsituting value of (i) in (ii)

⇒(4y+5)+(y)=80

⇒4y+5+y=80

⇒5y+5=80

⇒5y=80-5

⇒5y=75

⇒

⇒y=15

Question 3

**A girl was asked to multiply a numebr by . Instead she divided the numebr by and got the result 15 more than the correct result . What is the number ?**

Sol :

Let the number be x

On dividing x by

⇒

⇒

⇒

A.T.Q

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒x=56

Question 4

**What is the sum of two consecutive even numbers , the difference of whose square is 84 ?**

Sol :

Let the two consecutive even numbers be (2x) and (2x+2). Then ,

A.T.Q

⇒(2x+2)^{2}-(2x)^{2}=84

⇒{(2x)^{2}+2^{2}+2(2x)(2)}-4x^{2}=84 [using (a+b)^{2}=a^{2}+b^{2}+2ab]

⇒4x^{2}+4+8x-4x^{2}=84

⇒8x+4=84

⇒8x=84-4

⇒8x=80

⇒

⇒x=10

So , the numbers are 2x=2×10=20 and 2x+2=2×10+2=22

Then, sum is 20+22=42

Question 5

**One number is 7 more than another and its square is 77 more than the square of the smaller number. What are the numbers ?**

Sol :

Let the numbers are x and y.Then ,

A.T.Q

⇒x=y+7

⇒y=x-7..(i)

Also,

⇒(x)^{2}=77+y^{2}

⇒x^{2}=77+(x-7)^{2} [from (i)]

⇒x^{2}=77+x^{2}+49-2(x)(7) [(a-b)^{2}=a^{2}+b^{2}-2ab]

⇒x^{2}=77+x^{2}+49-14x

⇒x^{2}-x^{2}+14x=77+49

⇒14x=126

⇒

⇒x=9

Putting value of x in (i)

⇒y=9-7

⇒y=2

The numbers are 2 and 9

Question 6

**The numerator of a fraction is 4 less than its denominator if the numerator is decreased by 2 and the denominator is increased by 1 , then the fraction becomes . Find the fraction .**

Sol :

Let the numerator be x and denominator be y

A.T.Q

x=y-4..(i)

Also,

⇒

⇒8(x-2)=1(y+1)

⇒8x-16=y+1

⇒8x=y+1+16

⇒8x=y+17 ..(ii)

Subsituting (i) in (ii)

⇒8(y-4)=y+17

⇒8y-32=y+17

⇒7y=17+32

⇒

⇒y=7

Putting value of y in (i) , we get

⇒x=(7)-4

⇒x=3

Hence , the fraction is

Question 7

**The digits of a two-digit number are in the ratio 2:3 and the number obtained by interchanging the digits is greater than the original number by 27 . What is the original number ?**

Sol:

Let the common ration be x . Then , the tens digit be 2x and once digit be 3x .

Original number be 10(tens)+(once) which can be written as 10(2x)+3x=23x

Original number 23x ..(i)

On interchanging digits 10(3x)+2x=32x

And 32x is greater than original number by 27 , which means

⇒32x=23x+27

⇒32x-23x=27

⇒9x=27

⇒

⇒x=3

So , the tens digit be 2x=2(3)=6

The once digit(unit digit) be 3x=3(3)=9

Hence , the original number be 10(tens)+(once)=10(6)+9=60+9=69

Question 8

**The sum of the digits of a two- digit number is 10. The number formed by reversing the digits is 18 less than the original number. Find the original number .**

Sol :

Let the original number be 10x+y . Where **x** is tens and **y** is unit digi

A.T.Q

⇒x+y=10 [ sum of the digit of number ]

⇒x=10-y ..(i)

On reversing the number we get 10y+x which is equal to (10x+y)-18

⇒10y+x=(10x+y)-18

⇒10y+x=10x+y-18

⇒10y-y-10x+x=-18

⇒9y-9x=-18

⇒-9(-y+x)=-18

⇒

⇒x-y=2

⇒(10-y)-y=2

⇒10-y-y=2

⇒-2y=2-10

⇒-2y=-8

⇒

⇒y=4

Putting value of y in (i) , we get

⇒x=10-y

⇒x=10-4

⇒x=6

Hence , the original number be 10x+y=10(6)+4=64

Question 9.

**A man ate 100 grapes in 5 days. Each day he ate 6 more grapes than those he ate on the previous day. How many grapes did he eat the first day .**

Sol :

Let the grapes eaten at first day be **x . **Also , each day he ate 6 more grapes

A.T.Q

1st day + 2nd day + 3rd day + 4th day + 5th day = 100 grapes

⇒(x)+(x+6)+{(x+6)+6}+{(x+6+6)+6}+{(x+6+6+6)+6}=100

⇒x+(x+6)+(x+12)+(x+18)+(x+24)=100

⇒x+x+6+x+12+x+18+x+24=100

⇒5x+60=100

⇒5x=100-60

⇒5x=40

⇒

⇒x=8

Hence , grapes eaten by man on first day is 8

Question 10

**In an examination, a student scores 4 marks for every correct answer and losses 1 marks for every incorrect answer. If he attempts all 75 questions and secures 125 marks . What are the number of questions he attempted correctly ?**

Sol :

+4 Marks for every correct answer is given

-1 for every incorrect answer is given

Let the correct attempt be **x**

then , incorrect attempt be (75-x)

A.T.Q

⇒+4(x)-1(75-x)=125

⇒4x-75+x=125

⇒5x=125+75

⇒5x=200

⇒

⇒x=40

Which means 40 attempts are correct

Question 11

**Arun’s age is four years more than Prashant’s age 4 years ago , the ratio of their ages was 5 : 4 . Find their present ages.**

Sol:

Let the Prashant’s age be x (4 years ago from present)

Then Arun’s age be (x+4) (4 years ago from present)

A.T.Q

[cross multiply]

⇒4(x+4)=5x

⇒4x+16=5x

⇒5x-4x=16

⇒x=16

Prashant’s age 4 years ago be 16 And for present age (x+4) = 16+4 = 20 years

Arun’s age 4 years ago be (x+4) which is 20 and now at present {(x+4)+4} = {(16+4)+4}=24 years

Question 12

**A father is three times as old as his son. 15 years hence, he will be t****wice as old as his son. What is the sum of their present ages ?**

Sol :

Let the son age be x and father’s age be 3x

In 15 years

Son age be x + 15 and Father age be 2(x+15) “OR” 3x + 15 because both equation represent fathers present age

A.T.Q

⇒2(x + 15) = 3x + 15

⇒2x + 30 = 3x + 15

⇒30-15=3x-2x

⇒15=x

Therefore, Present age of son is x=15 and

fathers age is 3x = 3(15) = 45

And sum of their present age is 45+15 = 60 years

Question 13

**4 years ago, the ratio of their ages of A and B was 2 : 3 and after 4 years it becomes 5 : 7 . Find their present ages .**

Sol:

Given that, 4 years ago from present , the ratio of the ages of A and B = 2 : 3

Hence we can assume that

age of A 4 years ago = 2x

age of B 4 years ago = 3x

After 4 years from present, the ratio of their ages = 5 : 7

⇒{(2x+4)+4} **:** {(3x+4)+4} = 5 **:** 7

⇒

⇒7(2x+8)=5(3x+8) [cross multiplication]

⇒14x+56=15x+40

⇒56-40=15x-14x

⇒x=16

A’s present age = (2x+4) = 2(16)+4 = 36

B’s present age = (3x+4) = 3(16)+4 = 52

Question 14

**Length of a rectangular blackboards is 8 m more than its breadth. If its length is increased by 7 m and its breadth is decreased by 4 m , its area remains unchanged. What is the length of the blackboard ?**

Sol :

**Case 1:**

Let the breadth be x

Length of blackboard = x+8

Area of rectangular blackboard = l × b =(x) × (x+8)

= x^{2} + 8x..(i)

** **

**Case 2:**

Breadth is decreased by 4m then breadth = x-4

Length is increased by 7m then length = x+8+7 = x+15

Area of rectangular blackboard = l × b = (x-4) × (x+15)

= x(x+15) – 4(x+15)

= x^{2}+15x-4x-60

= x^{2}+11x-60..(ii)

Both (i) and (ii) are equal because it is given that area remains unchanged(equal)

⇒x^{2} + 8x = x^{2}+11x-60

⇒x^{2}-x^{2}+8x-11x=-60

⇒-3x=-60

⇒

⇒x=20

Length of blackboard = x+8 = 20+8 = 28

Question 15

**Two cars A and B leave Mumbai at the same time, travelling in opposite directions . If the average speed of car A is 8 km/h more than that of B, and they are 300 km apart at the end of 6 hours. What is the average speed of car A ?**

Sol :

Let x be average speed of B

Then Average speed of A = (x+8) km/hr

Distance travelled by A in 6 hrs = 6(x+8) km [Distance=speed×time]

Distance travelled by B in 6 hrs= 6(x) km

Distance travelled by A + Distance travelled by B = Total distance

⇒6(x-8)+6x=300

⇒6x-48+6x=300

⇒12x=300+48

⇒

⇒x=29

Hence , average speed of A is 29km/hrs

Question 16

**A boat takes 9 hours to travel a distance upstream and takes 3 hours to travel the same distance downstream. If the speed of the boat in still water is 4 km/h. What is the speed of the stream ?**

Sol :

Let the distance covered by the boat each way be **d** km. And let the velocity of the stream be **v** km/hr.

**Case 1 : (Downstream)**

While going downstream the speed of the boat is (v+4) km/hr and time taken is 3 hours

[Distance=velocity×time]

d=(+v+4)3..(i) [here , v is +ve because direction of stream and boat is same]

**Case 2 : (Upstream)**

While going upstream the speed of the boat is (v-4) km/hr and time taken is 9 hours

[Distance=velocity×time]

d=(-v-4)9..(ii) [here , v is -ve because direction of boat and stream is opposite]

(i)=(ii) , As the distance travelled upstream and downstream is the same,

⇒(v+4)3=(-v-4)9

⇒3v+12=-9v-36

⇒36+12=-9v-3v

⇒48=-12v

⇒

⇒-v=2km/hr [here , – minus sign shows direction of boat and stream]

or v = 2km/hr

Hence , the velocity of stream is 2km/hr

Question 17

**Kartik buys some apples at the rate of 10 rupees per apple . He also buys an equal number of oranges at the rate of 6 rupees per orange. He makes a 15% profit on apples and 8% profit on oranges. The total profit earned by Kartik at the end of the day is 396. When all fruit is sold out , find the number of apples purchased .**

Sol :

Number of apples be **x** and Number of oranges also be **x** (equal)

Cost of apples **10x **and cost of oranges be **6x**

⇒(15% of 10x )+(8% of 6x)=396

⇒

⇒

⇒

⇒198x=396×100

⇒

⇒x=200

Hence, Total number of apples purchased is 200

Question 18

**Bittu spends of the money with him on books , of the remaining on food and of the remaining on travel. Now , he is left with 450 rupees . How much money did he have in the beginning ?**

Sol :

Total money at the beginning be **x**

Spending on books [1/3 of total]

Remaining

Spending on foods [1/4 of remaining ] or

Remaining

Spending on travel , [2/5 of remaining]

Remaining

A.T.Q

Money left after spending on travel must be equal to 450

3x=450×10

x=1500

Hence , total money at beginning is 1500