**Exercise 3.9**

**1. A father is 3 times as old as his son. After 12 years, his age will be twice as that of his son then. Find their present ages.**

**Solution:**

Let the present age of the father = x years.

According to the question,

Father is 3 times as old as his son.

Thus, we get the equation,

x = 3y

\(\Rightarrow\) x â€“ 3y = 0

Similarly, according to the question,

After 12 years, fatherâ€™s age will be (x+12) years and sonâ€™s age will be (y+12) years.

We get the equation,

x + 12 = 2(y + 12)

\(\Rightarrow\) x + 12 = 2y + 24

\(\Rightarrow\) x â€“ 2y â€“ 12 = 0

The two equations are:

x â€“ 3y = 0

x – 2y – 12 = 0

Using cross-multiplication, we have

\(\frac{x}{(-3) \times (-12) – (-2) \times 0} = \frac{-y}{1 \times (-12) -1 \times 0} = \frac{1}{1 \times (-2) -1 \times (-3)}\)

\(\Rightarrow \frac{x}{36-0} = \frac{-y}{-12-0} \frac{1}{-2+3}\)

\(\frac{x}{36} = \frac{-y}{-12} = \frac{1}{1}\)

\(\frac{x}{36} = \frac{y}{12} = 1\)

x = 36, y = 12

Therefore, the present age of father is 36 years and the present age of son is 12 years.

**2. Ten years later. A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B.**

**Solution:Â **

Let the present age of A be x years

Let the present age of B be y years

According to the question,

After 10 years, Aâ€™s age will be (x +10) years and Bâ€™s age will be (y + 10) years.

Thus, we get the equation,

x â€“ 5 = 3(y – 5)

\(\Rightarrow\) x + 10 = 2y + 20

\(\Rightarrow\) x â€“ 2y -10 = 0

According to the question,

Before 5 years, the age of A was (x – 5) years and the age of B was (y – 5) years.

Thus, we get the equation,

x – 5 = 3(y-5)

\(\Rightarrow\) x – 5 = 3y – 15

\(\Rightarrow\) x – 3y + 10 = 0

The two equations are:

x – 2y – 10 = 0

x – 3y + 10 = 0

Using cross-multiplication, we get,

\(\frac{x}{(-2) \times 10 – (-3) \times (-10)} = \frac{-y}{1 \times 10 -1 \times (-10)} = \frac{1}{1 \times (-3) -1 \times (-2)}\)

\(\Rightarrow \frac{x}{-20-3-} = \frac{-y}{10+10} = \frac{1}{-3+2}\)

\(\Rightarrow \frac{x}{-50} = \frac{-y}{20} = \frac{1}{-1}\)

\(\Rightarrow \frac{x}{50} = \frac{y}{20} = 1\)

\(\Rightarrow x = 50, y=20\)

Therefore, the present age of A is 50 years and the present age if b is 20 years.

**3. A is elder to B by 2 years. Aâ€™s father F is twice as old as A and B is twice as old as his sister S. If the age of the father and sister differ by 40 years, find the age of A.**

**Solution:Â **

Let the present age of A = x

Let the present age of B = y

Let the present age of F = z

Let the present age of S = t

According to the question,

A is elder to b by 2 years. â‡’ x = y + 2

F is twice as old as A. â‡’ z = 2x

B is twice as old as S. â‡’ y = 2t

Given that the ages of F and S is differing by 40 years. â‡’ z – t = 40.

The four equations are:

x = y + 2,â€¦(i)

z = 2x,â€¦(ii)

y = 2t, â€¦(iii)

z – t = 40 â€¦(iv)

From the equations obtained, we know that,

x, y, z and t are unknowns.

We have to find the value of x.

So, by using equation(iii) in (i),

the first equation becomes x = 2t + 2

From the fourth equation, we have t = z – 40

Hence, we get

x = 2(z – 40) + 2

= 2z – 80 + 2

= 2z – 78

Using the equation (ii), we have

\(x = 2 \times 2x – 78\)

\(\Rightarrow x = 4x -78\)

\(\Rightarrow 4x -x = 78\)

\(\Rightarrow 3x = 78\)

\(\Rightarrow x = \frac{78}{3}\)

\(\Rightarrow x = 26\)

Therefore, the age of A is 26 years.

**4. Six year hence a manâ€™s age will be three times age of his son and three years ago he was nine times as old as his son. Find their present ages.**

**Solution:Â **

Let the present age of the man = x years

Let the present age of his son = y years.

According to the question,

After 6 years, the man’s age will be (x + 6) years and son’s age will be (y + 6) years.

Thus, we get the equation,

x+6=3(y+6)

x+6=3y+18

x-3y-12=0

According to the question,

Before 3 years, the age of the man was (x – 3) years and the age of son’s was (y â€” 3) years.

Thus, we get the equation,

x-3 = 9(y-3)

x-3 = 9y-27

x-9y+24=0

The two equations are,

x-3y-12=0

x-9y+ 24=0

Using cross-multiplication, we get

\(\frac{x}{(-3) \times 24 -(-9) \times (-12)} = \frac{-y}{1 \times 24 -1 \times (-12)} = \frac{1}{1 \times (-9) -1 \times (-3)}\)

\(\Rightarrow \frac{x}{-72-108} = \frac{-y}{24+12} = \frac{1}{9+3}\)

\(\Rightarrow \frac{x}{-180} = \frac{-y}{36} = \frac{1}{-9+3}\)

\(\Rightarrow \frac{x}{180} = \frac{y}{36} = \frac{1}{6}\)

\(\Rightarrow x= \frac{180}{6}, y=\frac{36}{6}\)

\(|\Rightarrow x = 30, y=6\)

Therefore, the present age of the man =30 years

And, the present age of son =6 years.

**5. Ten years ago, a father was 12 times as old as his son and 10 years hence, he will be twice as old as his son will be then. Find their present ages. **

**Solution:Â Â **

Let the present age of father =x years

Let the present age of his son =y years

According to the question,

After 10 years, father’s age will be (x+10) years and son’s age will be (y + 10) years.

Thus, we get the equation,

x+10=2(y+10)

x-10= 2y+20

x-2y-10=0

According to the question,

Before 10 years, the age of father was (x-10)years and the age of son was (y â€” 10) years.

Thus, we get the equation,

x-10 =12(y-10)

x-10 =12yâ€”I20

x-12y+110=0

The two equations are:

x-2y-10= 0

x-12y+110 = 0

Using cross-multiplication, we have

\(\frac{x}{(-2) \times 110 -(-12) \times (-10)} = \frac{-y}{1 \times 110 -1 \times (-10)} = \frac{1}{1 \times (-12) -1 \times (-12)}\)

\(\Rightarrow \frac{x}{-220-120} = \frac{-y}{110+10} = \frac{1}{-12+2}\)

\(\Rightarrow \frac{x}{-340} = \frac{-y}{120} = \frac{1}{-10}\)

\(\Rightarrow \frac{x}{340} = \frac{y}{120} = \frac{1}{10}\)

\(\Rightarrow x = \frac{340}{10} , y= \frac{120}{10} \)

\(\Rightarrow x=34, y=12\)

Therefore, the present age of father is 34 years and the present age of the son is 12 years.

**6. The present age of father is 3 years more than three times of the age of the son. Three years hence, fatherâ€™s age will be 10 years more than twice the age of the son. Determine their present age.**

**Solution:Â Â **

Let the present age of father =x years

Let the present age of his son =y years

According to the question,

The present age of father is three years more than three times the age of the son.

Thus, we get the equation,

x=3y+3 x-3y -3=0

According to the question,

After 3 years, father’s age will be (x + 3)years and son’s age will be (y + 3)years.

Thus, we get the equation,

x+3=2(y-F3)+10

x+3 = 4+6+10

x â€” 2y-13 = 0 So.

The two equations are:

x-3y-3= 0

x-2y-13 = 0

Using cross-multiplication, we get

\(\frac{x}{(-3) \times (-13) – (-2) \times (-3)} = \frac{-y}{1 \times (-13) -1 \times (-3)} = \frac{1}{1 \times (-2) -1 \times (-3)}\)

\(\Rightarrow \frac{x}{39-6} = \frac{-y}{-13+3} = \frac{1}{-2+3}\)

\(\Rightarrow \frac{x}{33} = \frac{-y}{-10} = \frac{1}{1}\)

\(\Rightarrow \frac{x}{33} = \frac{y}{10} = 1\)

\(\Rightarrow x = \frac{x}{33}, y= \frac{y}{10} \)

\(\Rightarrow x=33, y=10\)

Let the present age of father =33 years

Let the present age of his son =10 years

**7. A father is 3 times as old as his son. In 12 yearsâ€™ time, he will be twice as old as his son. Find the present ages of son and father.**

**Solution:Â Â **

Let the present age of father =x years

Let the present age of his son =y years.

According to the question,

The present age of father is three times the age of the son.

Thus, we get the equation,

x =3y

x-3y= 0

According to the question,

After 12 years, father’s age will be (x +12) years and son’s age will be (y +12)years.

Thus, we get the equation,

x+12 = 2(y+12)

x+12=2y+24

x-2y-12=0

The two equations are:

x-3y=0

x-2y-12=0

Using cross-multiplication, we get

\(\frac{x}{(-3) \times (-12) – (-2) \times 0} = \frac{-y}{1 \times (-12) -1 \times 0} = \frac{1}{1 \times (-2) -1 \times (-3)}\)

\(\Rightarrow \frac{x}{36-0} = \frac{-y}{-12-0} = \frac{1}{-2+3}\)

\(\Rightarrow \frac{x}{36} = \frac{-y}{-12} = \frac{1}{1}\)

\(\Rightarrow \frac{x}{36} = \frac{y}{12} = 1\)

\(\Rightarrow \frac{x}{33}= \frac{y}{10} \)

\(\Rightarrow x=36, y=12\)

Therefore, the present age of the father =36 years

And the present age of son =12 years.

**8. Fatherâ€™s age is three times the sum of age of his 2 children. After 5 years his age will be twice the sum of ages of two children. Find the age of father. **

**Solution:Â Â **

Let the present age of father =x years

Let the present ages of his two children’s be y and z years.

According to the question,

The present age of father is three times the sum of the ages of the two children’s.

Thus, we get the equation,

x =3(y+ z)

y+z= \(\frac{x}{3}\)

According to the question,

After 5 years, father’s age will be (x+5) years and the children’s age will be (y + 5) and (z +5) years.

Thus, we get the equation,

x+5 =2{(y+5)+(z+5)}

x+5= 2(y+5+z+5)

x = 2(y+z)+ 20 -5

x = 2(y + z)+I5

The two equations are:

y+z= \(\frac{x}{3}\)

x=2(y+z)+15

in the above equations, x, y and z are unknowns.

To find the value of x, substitute the value of (y+ z) from the equation(i) in the equation(ii),

By using cross-multiplication, we get,

x = \(\frac{2x}{3} + 15\)

\(\Rightarrow x – \frac{2x}{3} = 15\)

\(\Rightarrow x (1 – \frac{2}{3}) = 15\)

\(\Rightarrow \frac{x}{3} = 15\)

\(\Rightarrow x=15 \times 3\)

\(\Rightarrow x =45\)

Therefore, the present age of father is 45 years.

**9. Two years ago, a father was 5 times as old as his son. Two years later his age will be 8 more than 3 times the age of the son. Find the present ages of father and son.**

**Solution:Â Â **

Let the present age of father =x years

Let the present age of his son =y years

According to the question,

After 2 years, father’s age will be (x+ 2) years and the age of son will be (y + 2) years.

Thus, we get the equation,

x+2=3(y+2)+8

x+2=3y+6+8

x-3y-12=0

According to the question,

Before 2 years, the age of father was (x â€”2) years and the age of son was (yâ€” 2) years.

Thus, we get the equation,

x-2 =5(y-2)

x-2 =5y-10

x-5y+8=0

The two equations are:

x – 3y -12 = 0

x – 5y +8 = 0

Using cross-multiplication, we get,

\(\frac{x}{(-3) \times 8 – (-5) \times (-12)} = \frac{-y}{1 \times 8 -1 \times (-12)} = \frac{1}{1 \times (-5) -1 \times (-3)}\)

\(\Rightarrow \frac{x}{-24-60} = \frac{-y}{8+12} = \frac{1}{-5+3}\)

\(\Rightarrow \frac{x}{-84} = \frac{-y}{20} = \frac{1}{-2}\)

\(\Rightarrow \frac{x}{84} = \frac{y}{20} = \frac{1}{2}\)

\(\Rightarrow x = \frac{84}{2}, y= \frac{20}{2} \)

\(\Rightarrow x=42, y=10\)

Therefore, the present age of father = 42 years

And, the present age of son =10 years.

**10. Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?**

**Solution:Â Â **

Let the present age of Nuri =x years

Let the present age of Sonu =y years.

According to the question,

After 10 years, Nuri’s age will be(x + 10) years and the age of Sonu will be(y + 10) years.

Thus, we get the equation,

x +10 =2(y+10)

x + 10 = 2y + 20

x – 2y – 10 =0

According to the question,

Before 5 years, the age of Nuri was (x – 5)years and the age of Sonu was (y- 5)years.

Thus, we get the equation,

x -5 =3(y-5)

x – 5 =3y-15

x-3y+10=0

The two equations are:

x – 2y – 10 = 0

x-3y+10=0

Using cross-multiplication, we get,

\(\frac{x}{(-2) \times 10 – (-3) \times (-10)} = \frac{-y}{1 \times 10 -1 \times (-10)} = \frac{1}{1 \times (-3) -1 \times (-2)}\)

\(\Rightarrow \frac{x}{-20-30} = \frac{-y}{10+10} = \frac{1}{-3+2}\)

\(\Rightarrow \frac{x}{-50} = \frac{-y}{20} = \frac{1}{-1}\)

\(\Rightarrow \frac{x}{50} = \frac{y}{20} = 1\)

\(\Rightarrow x=50, y=20\)

Let the present age of Nuri =50Â years

Let the present age of Sonu =20Â years.

**11. The ages of two friends Ani and biju differ by 3 years. Aniâ€™s father Dharam is twice as old as Ani and Biju as twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.**

**Solution:Â Â **

Let the present ages of Ani = x years

Let the present ages of Biju= y years

Let the present ages of Dharam = z years

Let the present ages of Cathy = t years

According to the question,

The ages of Ani and Biju differ by 3 years.

Thus, we get the equation,

x – y=Â±3

x=yÂ±3

According to the question,

Dharam is twice as old as Ani.

Thus, we get the equation,

z = 2x

According to the question,

Biju is twice as old as Cathy.

Thus, we get the equation,

y = 2t

According to the question,

The ages of Cathy and Dharam differ by 30 years. Clearly, Dharam is older than Cathy.

Thus, we get the equation,

z – t =30

The two systems of simultaneous equations are:

(i) Â x=y+3,

z = 2x,

y=2t,

z- t =30

(ii) x = y-3,

z = 2x,

y=2t,

z – t=30

Since, x, y, z and t are unknowns.

We have to find the value of x and y.

(i) By using the third equation, the first equation becomes x = 2t +3

From the fourth equation, we have

t = z -30

Hence, we have

x = 2(z-30)+3

=2z-60+3

=2z-57

Using the second equation, we have

\(x = 2 \times 2x -57\)

x = 4x-57

4x – x = 57

3x =57

\(x = \frac{57}{3}\)

x = 19

From the first equation, we have

x =y+3

y=x-3

y=19-3

y = 16

Hence, the age of Ani is 19 years and the age of Biju is 16 years.

(ii) By using the third equation, the first equation becomes x= 2t -3.

From the fourth equation, we have

tÂ = z – 30

Hence, we have

x = 2(z â€”30)-3

=2z-60-3

= 2z-63

Using the second equation, we have

\(x = 2 \times 2x – 63\)

x = 4x-63

4x – x = 63

3x =63

\(x = \frac{63}{3}\)

x = 21

From the first equation, we have

x = y-3

y= x + 3

y= 21 + 3

y= 24

Therefore, the age of Ani =21 years

And, the age of Biju is 24 years.

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