**Exercise 8.7**

**Question 1:Â ****Find the consecutive numbers whose squares have the same sum of 85.**

**Solution:**

Consider two consecutive natural numbers as x and x+1.

It is given that that the sum of their squares is 85.

So,Â x^{2 }+(x+1)^{2 }=85

= x^{2}+x^{2}+2x+1 =85

= 2x^{2}+ 2x+ -84 =0

= 2(x^{2}+ x+ -42) =0

=Â x^{2}+ x+ -42 = 0

Now, the roots of this equation can be found using factorization method

So, x^{2}+ 7x-6x -42 =0

= x(x+7) -6(x+7) =0

= (x – 6) (x + 7) = 0

So, x = 6 Or, x = -7

And, x + 1 = 7 Or -6

Thus, the consecutive numbers whose squares have the same sum of 85 are (6, 7) or (-7, -6).

**Question 2:Â ****Divide 29 into two parts so that the sum of the squares of the parts is 425.**

**Solution:**

Consider 29 is divided into two parts as x and 29-x.

As the sum of the two parts is 425,

â‡’Â x^{2} +(29-x)^{2} = 425

= x^{2}+x^{2}+841+(-58x) = 425

= 2x^{2}-58x+416 =0

= x^{2}-29x +208 =0

This equation is now in quadratic form and can be solved using factorization method

So, x^{2 }-13x-16x+208 =0

= x(x-13) -16(x-13) =0

= (x-13)(x-16) =0

Here, x=13 Or,Â x = 16

Hence, 29 can be divided into 13 and 16 which follows the condition (13^{2} + 16^{2} = 425).

**Question 3:Â ****Two squares have sides x cm and (x+4) cm. The sum of their areas is 656 cm ^{2}.find the sides of the squares.**

**Solution:**

Area of a square = sideÂ Ã— side

From the given parameters, area of the square = x (x + 4) cm^{2}

Also, it is given that the sum of the areas is 656 cm^{2}

So,

= x(x+4) +x(x+4) = 656

= 2x(x+4) = 656

= x^{2 }+4x =328

Using factorization method to find “x”-

= x^{2 }+20x-16x-328 =0

= x(x+20)-16(x+20) =0

= (x+20)(x-16) =0

â‡’ x = -20 Or, x= 16

Since the value of a square cannot be negative, x can only be a positve value.

So, the side of the square is 16.

And, x+4 =16+4 =20 cm

Thus, the side of the square is 20 cm.

**Question 4:Â ****The sum of two numbers is 48 and their product is 432. Find the numbers.**

**Solution:**

Consider two number as x and 48-x.

Given,

=x(48-x) = 432

= 48x-x^{2}=432

= x^{2}-48x+432=0

= x^{2}-36x-12x+432=0

= x(x-36)-12(x-36) =0

=(x-36)(x-12) =0

â‡’ x = 36Â Or, x = 12

Thus, 12 and 36 are the required two numbers.

**Question 5:Â ****If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.**

**Solution:**

Consider an integer x which whenÂ added to its square, the sum becomes 90

So, x+ x^{2} = 90

= x^{2} +x- 90=0

= x^{2} +10x-9x- 90=0

= x(x+10)-9(x+10) =0

= (x+10)(x-9) =0

â‡’ x= -10 Or,Â x =9

Thus, the values of the integer are 9 and -10.

**Question 6:Â ****Find the whole numbers which when decreased by 20 is equal to 69 times the reciprocal of the numbers.**

**Solution:**

Consider any whole number as x cm

As it is decreased by 20 = (x-20) = \(\frac{69}{x}\)

x-20 =\(\frac{69}{x}\)

x(x-20) =69

x^{2 }Â -20x -69 =0

Using factorization method,

x^{2} – 23x + 3x – 69 = 0

x(x-23) +3(x-23)=0

(x-23)(x+3) =0

So, x= 23 Or, x= -3

It is known that whole numbers are always positive and so, x= -3 is not considered.

âˆ´ The required whole number is 23.

**Question 7:Â ****Find the consecutive natural numbers whose product is 20.**

**Solution:**

Consider two consecutive natural numbers as x and x+1.

It is given that the product of these two natural numbers is 20

So, x(x+1) =20

= x^{2}+x-20 =0

= x^{2}+5x-4x-20 =0

= x(x+5)-4(x+5) =0

= (x+5)(x-4) =0

Now, x =-5 Or,Â x =4

So, the two consecutive natural numbers whose product is 20 are (4, 5) and (-5, -4).

**Question 8:Â ****The sum of the squares of two consecutive odd positive integers is 394. Find the two numbers?**

**Solution:**

Consider two consecutive odd positive integers as 2x-1 and 2x+1 respectively.

It is given that their sum of squares is 394.

So,

(2x-1)^{2}+ (2x+1)^{2 }= 394

4x^{2 }+1-4x+4x^{2}+1+4x = 394

â‡’ 8x^{2}+ 2 = 394

8x^{2} = 392

x^{2} = 49

x = 7 and -7

Here, x = -7 cannot be considered as the value of the edge of the square cannot be negative.

Now, 2x-1 = 14-1 = 13

2x+1 = 14 +1 = 15

Thus, the consecutive odd positive numbers are 13 and 15 respectively.

**Question 9:Â ****The sum of two numbers is 8 and 15 times the sum of the reciprocal is also 8 . Find the numbers.**

**Solution:**

Let the numbers be x and 8-x respectively.

Given that the sum of the numbers is 8 and 15 times the sum of their reciprocals.

So,

= \(15(\frac{1}{x}+\frac{1}{8-x})=8\)

= \(15\frac{8-x+x}{x(8-x)}=8\)

= \(15\times \frac{8}{8x-x^{2}}=8\)

= 120=8(8x-x^{2})

= 120 = 64x-8x^{2}

=8x^{2}-64x+120=0

= 8(x^{2}-8x+15)=0

= x^{2}-8x+15=0

= x^{2}-5x-3x+15=0

=x(x-5)-3(x-5) =0

= (x-5)(x-3) =0

Hence, x =5 Or, x =3

Thus, The two numbers are 5 and 3 respectively.

**Question 10:Â ****The sum of a number and its positive square root is \(\frac{6}{25}\). Find the numbers.**

**Solution:**

Consider the number to be x

From question,

\(x+\sqrt{x}=\frac{6}{25}\)Now, assume that x=y^{2} ,

\(y+y^{2}=\frac{6}{25}\)

= 25y^{2}+25y-6 =0

The value of y can be determined by:

\(y=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

Where a = 25, b = 25 , c =-6

\(y=\frac{-25\pm \sqrt{625+600}}{50}\)

\(y=\frac{-25\pm 35}{50}\)

\(y=\frac{1}{5}\,and\,y=\frac{-11}{10}\)

= x=y^{2}= \(\frac{1}{5}^{2}=\frac{1}{25}\)

The number x is \(\frac{1}{25}\)

**Question 11:Â ****There are three consecutive integers such that the square of the first increased by the product of the other two integers gives 154.Â What are the integers?**

**Solution:**

Let the three consecutive numbers be x, x+1, x+2 respectively.

X^{2}+(x+1)(x+2) =154

= x^{2}+x^{2}+3x+2 =154

= 3x^{2}+3x-152=0

The value of x can be obtained by the formula

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

Here a =3 , b =3 , c =152

x=Â \(x=\frac{-3\pm \sqrt{9-1216}}{6}\)

\(x=8\,and\,x=\frac{-19}{2}\)

Considering the value of x

If x=8

x+1 =9

x+2 = 10

The three consecutive numbers are 8 , 9 , 10 respectively.

**Question 12:Â ****The product of two successive integral multiples of 5 is 300. Determine the multiples.**

**Solution:**

Given that the product of two successive integral multiples of 5 is 300

Let the integers be 5x and 5(x+1)

According to the question,

5x[5(x+1)] = 300

= 25x(x+1) =300

= x^{2}+x =12

= x^{2}+x -12=0

= x^{2}+4x-3x -12=0

= x(x+4)-3(x+4) =0

=(x+4)(x-3) =0

Either x+4 =0

Therefore x=-4

Or, x-3 =0

Therefore x =3

x =-4

5x = -20

5(x+1) = -15

x=3

5x =15

5(x+1) = 20

The two successive integral multiples are 15,20 and -15 and -20 respectively.

**Question 13:Â ****The sum of the squares of two numbers is 233 and one of the numbers is 3 less than the other number. Find the numbers.**

**Solution:**

Let the number is x

Then the other number is 2x-3

According to the question:

x^{2}+(2x-3)^{2} = 233

= x^{2}+4x^{2}+9-12x = 233

= 5x^{2}-12x-224 =0

The value of x can be obtained by \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

Here a= 5 , b= -12 , c =-224

x = \(x=\frac{12\pm \sqrt{144+20(224)}}{2(5)}\)

\(x=8\,and\,x=\frac{-28}{5}\)

Considering the value of x =8

2x-3 = 15

The two numbers are 8 and 15 respectively.

**Question 14:Â ****The difference of two number is 4 . If the difference of the reciprocal is \(\frac{4}{21}\) . find the numbers.**

**Solution:**

Lethe two numbers be x and x-4 respectively.

Given, that the difference of two numbers is 4 .

By the given hypothesis we have,

=\(\frac{1}{x-4}-\frac{1}{x}=\frac{4}{21}\)

=\(\frac{x-x+4}{x(x-4)}=\frac{4}{21}\)

= 84 = 4x(x-4)

= x^{2}-4x-21 =0

Applying factorization theorem,

= x^{2 }-7x+3x-21 =0

=(x-7)(x+3) =0

Either x-7 =0 therefore x= 7

Or, x+3 = 0 therefore x = -3

Hence the required numbers are -3 and 7 respectively.

**Question 15:Â ****Let us find two natural numbers which differ by 3 and whose squares have the sum 117.**

**Solution:**

Let the numbers be x and x-3

According to the question

x^{2}+(x-3)^{2}=117

= x^{2}+x^{2}+9-6x-117 =0

= 2x^{2}-6x-108 =0

= x^{2}-3x-54 =0

= x^{2}-9x+6x-54 =0

= x(x-9)+6(x-9) =0

=(x-9)(x+6) =0

Either x-9 =0 therefore x=9

Or ,x+6 =0 therefore x=-6

Considering the positive value of x that is 9

x=9

x-3 = 6

The two numbers are 6 and 9 respectively.

**Question 16:Â ****The sum of the squares of these consecutive natural numbers is 149. Find the numbers.**

**Solution:**

Let the numbers be x , x+1, and x+2 respectively.

According to given hypothesis

X^{2}+ (x+1)^{2}+(x+2)^{2} =149

X^{2}+ X^{2} + X^{2 }+1+2x+4+4x = 149

3x^{2} +6x-144 =0

X^{2}+2x-48=0

Now applying factorization method,

X^{2 }+8x-6x-48=0

X(x+8)-6(x+8) =0

(x+8)(x-6) =0

Either x+8 =0 therefore x= -8

Or, x-6 =0 therefore x= 6

Considering only the positive value of x that is 6 and discarding the negative value.

x=6

x+1 = 7

x+2 = 8

The three consecutive numbers are 6 , 7 , and 8 respectively.

**Â **

**Question 17:Â ****Sum of two numbers is 16. The sum of their reciprocal is \(\frac{1}{3}\).find the numbers.**

**Solution:**

Given that the sum of the two natural numbers is 16

Let the two natural numbers be x and 16-x respectively

According to the question

= \(\frac{1}{x}+\frac{1}{16-x}=\frac{1}{3}\)

= \(\frac{16-x+x}{x(16-x)}=\frac{1}{3}\)

= \(\frac{16}{x(16-x)}=\frac{1}{3}\)

= 16x-x^{2} =48

= -16x+x^{2}+48 =0

= +x^{2}-16x+48=0

= +x^{2}-12x-4x+48=0

= x(x-12)-4(x-12) =0

= (x-12)(x-4) =0

Either x-12 =0 therefore x= 12

Or , x-4 =0 therefore x= 4

The two numbers are 4 and 12 respectively.

**Question 18:Â ****Determine the two consecutive multiples of 3 whose product is 270**

**Solution:**

Let the consecutive multiples of 3 are 3xand 3x+3

According to the question

3x(3x+3) = 270

= x(3x+3) =90

= 3x^{2}+3x =90

= 3x^{2}+3x -90=0

= x^{2}+x -30=0

= x^{2}+6x-5x -30=0

=x(x+6)-5(x+6) =0

= (x+6)(x-5) =0

Either x+6 = 0 therefore x=-6

Or , x-5 = 0 therefore x=5

Considering the positive value of x

x=5

3x = 15

3x+3 = 18

The two consecutive multiples of 3 are 15 and 18 respectively.

**Question 19:Â ****The sum of a number and its reciprocal is \(\frac{17}{4}\) . find the numbers.**

**Solution:**

Lethe number be x

According to the question

\(\frac{x^{2}+1}{x}=\frac{17}{4}\)

= 4(x^{2}+1)=17x

= 4x^{2}+4-17x=0

= 4x^{2}+4-16x-x=0

= 4x(x-4)-1(x-4) =0

=(4x-1)(x-4) =0

Either x-4 =0 therefore x=4

Or, 4x-1 =0 therefore \(x=\frac{1}{4}\)

The value of x is 4

**Question 20:Â ****A two digit is such that the products of its digits is 8when 18 is subtracted from the number, the digits interchange their places. Find the number?**

**Solution:**

Let the digits be x and x-2 respectively.

The product of the digits is 8

According to the question

x(x-2) = 8

= x^{2}-2x-8 =0

= x^{2}-4x+2x-8 =0

= x(x-4)+2(x-4) =0

Either x-4 =0 therefore x=4

Or , x+2 =0 therefore x= -2

Considering the positive value of x = 4

x-2 = 2

The two digit number is 42.

**Question 21:Â ****A two digit number is such that the product of the digits is 12, when 36 is added to the number, the digits interchange their places .find the number.**

**Solution:**

Let the tens digit be x

Then, the unit digit = \(\frac{12}{x}\)

Therefore the number = \(10x+\frac{12}{x}\)

And, the number obtained by interchanging the digits = \(x+\frac{120}{x}\)

= \(10x+\frac{12}{x}+36=x+\frac{120}{x}\)

= \(9x+\frac{12-120}{x}+36=0\)

= \(\frac{9x^{2}+{12-120}{x}+36x}{x}=0\)

= \(\frac{9x^{2}+{-108}{x}+36x}{x}=0\)

= 9(x^{2}+4x-12)=0

= (x^{2}+4x-12)=0

= x^{2}+6x-2x-12=0

= x(x+6)-2(x+6) =0

=(x-2)(x+6) =0

Either x-2 = 0 therefore x=2

Or, x+6 =0 therefore x= -6

Since a digit can never be negative. So x=2

The number is 26.

**Question 22:Â ****A two digit number is such that the product of the digits is 16 when 54 is subtracted from the number, the digits are interchanged. Find the number.**

**Solution:**

Let the two digits be:

Tens digit be x

Units digit be \(\frac{16}{x}\)

Numbers = \(10x+\frac{16}{x}\) â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(i)

Number obtained by interchanging = \(10(10x+\frac{16}{x})\)

\(10x+\frac{16}{x}\) – \(10(10x+\frac{16}{x})\) =54

= 10x^{2}+16-160+x^{2} = 54

= 9x^{2}-54x-144= 0

= x^{2}-6x-16 =0

= x^{2}-8x+2x-16 =0

= x(x-8)+2(x-8) =0

=(x-8)(x+2)=0

Either x-8 =0 therefore x=8

Or, x+2 =0 therefore x =-2

A digit can never be negative so x = 8

Hence by putting the value of x in the above equation (i) the number is 82.

**Question 23:Â ****Two numbers differ by 3 and their product is 504. Find the numbers.**

**Solution:**

Let the numbers be x and x-3 respectively.

According to the question

= x(x-3) =504

=x^{2}-3x-504 =0

= x^{2}-24x+21x-504 =0

= x(x-24)+21 (x-24) =0

=(x-24)(x+21) =0

Either x-24 = 0 therefore x =24

Or , x+21 =0 , therefore x =-21

x = 24 and x= -21

x-3 = 21 and x-3 = -24

The two numbers are 21 a nd 24 and -21 and -24 respectively.

**Question 24:Â ****Two numbers differ by 4 and their product is 192. Find the numbers.**

**Solution:**

Let the two numbers be x and x-4 respectively

Given that the product of the numbers is 192

According to the question

= x(x-4) = 192

= x^{2}-4x -192 =0

=Â Â x^{2}-16x+12x -192 =0

= x(x-16) +12(x-16) =0

= (x-16) (x+12) =0

Either x-16 =0 therefore x= 16

Or, x+12 =0 therefore x= -12

Considering only the positive value of x

x=16S

x-4 = 12

The two numbers are 12 and 16 respectively.

**Question 25:Â ****A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the numbers.**

**Solution:**

Let the digit in the tens and the units place be x and y respectively.

Then it is represented by 10x+y

According to the question,

10x+y = 4(sum of the digits) and 2xy

10x+y = 4(x+y) and 10x+y = 2xy

10x+y = 4x+4y and 10x+y = 2xy

6x-3y =0 and 10x+y-2xy =0

y= 2x and 10x + 2x -2x(2x) = 0

12x= 4x^{2 }

4x(x-3) =0

Either 4x=0 therefore x= 0

Or, x-3 =0 therefore x= 3

We have y = 2x

When x= 3 , y= 6

**Question 26:Â ****The sum of the squares of two positive integers is 208. If the square of the large number is 18 times the smaller. Find the numbers.**

**Solution:**

Let the smaller number be x

Then, square of the large number be = 18x

Also, square of the smaller number be = x^{2}

It is given that the sum of the square of the integer is 208.

Therefore,

= x^{2} + 18x =208

= x^{2}+18x -208 =0

Applying factorization theorem,

= x^{2} +26x-8x-208 =0

= x(x+26)-8(x+26) =0

= (x+26)(x-8) =0

Either x+26=0 therefore x=-26

Or, x-8 =0 therefore x= 8

Considering the positive number, therefore x= 8.

Square of the largest number =18x = 18*8 = 144

Largest number = \(\sqrt{144} = 12\)

Hence the numbers are 8 and 12 respectively.

**Question 27:Â ****The sum of two numbers is 18. The sum of their reciprocal is \(\frac{1}{4}\) .find the numbers.**

**Solution:**

Let the numbers be x and (18-x) respectively.

According to the given hypothesis,

\(\frac{1}{x}-\frac{1}{18-x} = \frac{1}{4}\)

\(\frac{18-x+x}{x(18-x)}= \frac{1}{4}\)

\(\frac{18}{-x^{2}+18x}= \frac{1}{4}\)

= 72 = 18x-x^{2}

= x^{2}-18x+72 =0

Applying factorization theorem, we get,

= x^{2} -6x-12x +72=0

= x(x-6)-12(x-6) =0

= (x-6)(x-12) =0

Either, x= 6

Or, x=12

The two numbers are 6 and 12 respectively.

**Â **

**Question 28:Â ****The sum of two numbers a and b is 15 and the sum of their reciprocals \(\frac{1}{a}\) and \(\frac{1}{b}\) is \(\frac{3}{10}\). Find the numbersÂ a and b.**

**Solution:**

Let us assume a number x such that

\(\frac{1}{x}-\frac{1}{15-x}=\frac{3}{10}\)

\(\frac{15-x+x}{x(15-x)}=\frac{3}{10}\)

\(\frac{15}{15x-x^{2}}=\frac{3}{10}\)

= 3x^{2}-45x+150=0

= x^{2} -15 x+50 = 0

Applying factorization theorem,

= x^{2}– 10x-5x+50=0

= x(x-10)-5(x-10) =0

= (x-10)(x-5) =0

Either, x-10 =0 therefore x=10

Or, x-5=0 therefore x=5

Case (i)

If x = a , a=5 and b= 15-x , b= 10

Case (ii)

If x = 15-a = 15-10 = 5 ,

x=a=10 , b= 15-10 =5

Hence when a=5 , b=10

a=10 , b= 5

**Â ****Question 29:Â ****The sum of two numbers is 9. The sum of their reciprocal is \(\frac{1}{2}\).find the numbers.**

**Solution:**

Given that the sum of the two numbers is 9

Let the two number be x and 9-x respectively

According to the question

\(\frac{1}{x}+\frac{1}{9-x}=\frac{1}{2}\)

= \(\frac{9-x+x}{x(9-x)}=\frac{1}{2}\)

= \(\frac{9}{9x-x^{2}}=\frac{1}{2}\)

= 9x-x^{2}= 18

= x^{2}-9x+18 =0

= x^{2}-6x-3x+18 =0

= x(x-6)-3(x-6) =0

= (x-6)(x-3)=0

Either x-6 =0 therefore x= 6

Or x-3 =0 therefore x=3

The two numbers are 3 and 6 respectively

**Question 30:Â ****Three consecutive positive integers are such that the sum of the squares of the first and the product of the other two is 46. Find the integers.**

**Solution:**

Let the consecutive positive integers be x , x+1, x+2 respectively

According to the question

X^{2}+(x+1)(x+2) = 46

= x^{2}+x^{2}+3x+2 = 46

=2 x^{2}+3x+2 = 46

= 2 x^{2}+3x+2 -46=0

= 2 x^{2}-8x+11x+ -44=0

= 2x(x-4)+11(x-4) =0

= (x-4)(2x+11) =0

Either x-4 =0 therefore x=4

Or, 2x+11 =0 thereforeÂ \(x=\frac{-11}{2}\)

Considering the positive value of x that is x= 4

The three consecutive numbers are 4 , 5Â and 6 respectively

**Question 31:Â ****The difference of squares of two numbers is 88. If the large number is 5 less than the twice of the smaller, then find the two numbers**

**Solution:**

Let the smaller number be x and larger number is 2x-5

It is given that the difference of the squares of the number is 88

According to the question

(2x-5)^{2}-x^{2}= 88

= 4x^{2}+25-20x- x^{2} =88

= 3x^{2}-20x-63 =0

= 3x^{2}-27x+7x-63 =0

= 3x(x-9)+7(x-9) =0

= (x-9)(3x+7)=0

Either x-9 =0 therefore x=9

Or, 3x+7 =0 therefore \(x=\frac{-7}{3}\)

Since a digit can never be negative so x= 9

Hence the number is 2x-5 = 13

The required numbers are 9 and 13 respectively

**Question 32:Â ****The difference of squares of two numbers is180. The square of the smaller number is 8 times the larger number. Find the two numbers**

**Solution:**

Let the number be x

According to the question

X^{2}-8x = 180

X^{2}-8x-180 =0

= X^{2}+10x-18x-180 =0

= x(x+10)-18(x-10) =0

= (x-18)(x+10) =0

Either x-18 = 0 therefore x= 18

Or, x+10 =0 therefore x=-10

Case (i)

X=18

8x= 144

Larger number = \(\sqrt{144}=12\)

Case (ii)

X= -10

Square of the larger number 8x= -80

Here in this case no perfect square exist

Hence the numbers are 18 and 12 respectively .