Pair of Linear Equations: Word Problems

Worked example: an age problem with two cases

Ani and Biju's ages differ by 3 years. Ani's father Dharam is twice as old as Ani, and Biju is twice as old as his sister Cathy. Cathy and Dharam's ages differ by 30 years. Find Ani and Biju's ages.

Solution: Let Ani's age be $x$ and Biju's be $y$ . "Differ by 3 years" doesn't say who's older, so there are genuinely two cases to check.

Case 1: Ani is older. Then $x-y=3$ . Dharam's age is $2x$ , Cathy's is $\tfrac{y}{2}$ , and their ages differ by 30:

2x - \dfrac{y}{2} = 30 \;\Rightarrow\; 4x-y=60

Subtracting $x-y=3$ from this:

3x = 57 \;\Rightarrow\; x=19, \quad y = 19-3=16

Case 2: Biju is older. Then $y-x=3$ , and the same age-difference equation $4x-y=60$ still holds. Adding the two equations this time:

3x = 63 \;\Rightarrow\; x=21, \quad y=21+3=24

Both $(19,16)$ and $(21,24)$ are valid, the problem's wording alone doesn't rule either out.

Worked example: a classic money problem

Two friends have some money each. The first says: "Give me ₹100 and I'll have twice what you have left." The second says: "Give me ₹10 instead and I'll have six times what you have left." Find how much each friend has.

Solution: Let the first friend have $\text{₹}x$ and the second have $\text{₹}y$ .

First condition (second friend gives ₹100 to the first):

x+100 = 2(y-100) \;\Rightarrow\; x-2y=-300

Second condition (first friend gives ₹10 to the second):

y+10 = 6(x-10) \;\Rightarrow\; 6x-y=70

Multiplying the second equation by 2 and subtracting the first:

12x-2y=140, \qquad (12x-2y)-(x-2y) = 140-(-300) \;\Rightarrow\; 11x=440 \;\Rightarrow\; x=40

Substituting back: $40-2y=-300 \;\Rightarrow\; y=170$ . Checking the first condition: $40+100=140$ and $2(170-100)=140$ . Matches.

Pair of Linear Equations: Word Problems

Worked example: an age problem with two cases

Ani and Biju's ages differ by 3 years. Ani's father Dharam is twice as old as Ani, and Biju is twice as old as his sister Cathy. Cathy and Dharam's ages differ by 30 years. Find Ani and Biju's ages.

Solution: Let Ani's age be $x$ and Biju's be $y$ . "Differ by 3 years" doesn't say who's older, so there are genuinely two cases to check.

Case 1: Ani is older. Then $x-y=3$ . Dharam's age is $2x$ , Cathy's is $\tfrac{y}{2}$ , and their ages differ by 30:

2x - \dfrac{y}{2} = 30 \;\Rightarrow\; 4x-y=60

Subtracting $x-y=3$ from this:

3x = 57 \;\Rightarrow\; x=19, \quad y = 19-3=16

Case 2: Biju is older. Then $y-x=3$ , and the same age-difference equation $4x-y=60$ still holds. Adding the two equations this time:

3x = 63 \;\Rightarrow\; x=21, \quad y=21+3=24

Both $(19,16)$ and $(21,24)$ are valid, the problem's wording alone doesn't rule either out.

Worked example: a classic money problem

Solution: Let the first friend have $\text{₹}x$ and the second have $\text{₹}y$ .

First condition (second friend gives ₹100 to the first):

x+100 = 2(y-100) \;\Rightarrow\; x-2y=-300

Second condition (first friend gives ₹10 to the second):

y+10 = 6(x-10) \;\Rightarrow\; 6x-y=70

Multiplying the second equation by 2 and subtracting the first:

12x-2y=140, \qquad (12x-2y)-(x-2y) = 140-(-300) \;\Rightarrow\; 11x=440 \;\Rightarrow\; x=40

Substituting back: $40-2y=-300 \;\Rightarrow\; y=170$ . Checking the first condition: $40+100=140$ and $2(170-100)=140$ . Matches.

Pair of Linear Equations: Word Problems

Worked example: an age problem with two cases

Worked example: a classic money problem

Continue learning

Pair of Linear Equations: Word Problems

Worked example: an age problem with two cases

Worked example: a classic money problem

Continue learning