Q) At present, Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the sum of their present ages.
Ans:
METHOD 1 (ONE VARIABLE)
Let’s consider the Nisha’s age is Y.
Step 1: By 1st condition: “Asha’s age is 2 more than the square of her daughter Nisha’s age”
∴ Asha’s age = Y 2 + 2 ……. (i)
Step 2: Time gap for 2nd condition: “When Nisha grows to her mother’s present age”
Time gap = Asha’s present age – Nisha’s present age
= ( Y 2 + 2) – Y
= (Y 2 – Y + 2) years
Now, Asha’s age after (Y 2 – Y + 2) years = (Y 2 + 2) + (Y 2 – Y + 2) = (2 Y 2 – Y + 4) years
and Nisha’s age after (Y 2 – Y + 2) years = Y + (Y 2 – Y + 2) = (Y 2 + 2) (it meets the given condition)
Step 3: By 2nd condition (after this time gap): “Asha’s age would be one year less than 10 times the present age of Nisha.”
∴ (2 Y 2 – Y + 4) = 10 Y – 1
∴ 2 Y 2 – 11 Y + 5 = 0
∴ 2 Y 2 – 10 Y – Y + 5 = 0
∴ 2 Y (Y – 5) – (Y – 5) = 0
∴ (Y – 5) (2 Y – 1) = 0
∴ Y = 5 and 1/2 years
Since, age can not be in fraction value, hence we reject Y = 1/2 and accept Y = 5
∴ Y = 5 years (Nisha’s age)
Step 4: By substituting value of Y in equation (ii), we get:
∵ 2 X = 11 Y – 1
∴ 2 X = 11 (5) – 1 = 54
∴ X = 27 years
Step 5: Sum of their present ages = Asha’s present age + Nisha’s present age
= 27 + 5 = 32 years
Therefore, the sum of their present ages is 32 years.
METHOD 2 (TWO VARIABLES)
Let’s consider the Asha’s age is X and Nisha’s age is Y.
Step 6: By 1st condition: “Asha’s age is 2 more than the square of her daughter Nisha’s age”
∴ X = Y 2 + 2 ……. (i)
Step 7: Time for 2nd condition: “When Nisha grows to her mother’s present age”
Time gap = X – Y
∴ Asha’s age after (X – Y) years = X + ( X – Y ) = 2 X – Y
∴ Nisha’s age after (X – Y) years = Y + ( X – Y ) = X (it meets the given condition)
Step 8: By 2nd condition (after this time gap): “Asha’s age would be one year less than 10 times the present age of Nisha.”
∴ 2 X – Y = 10 Y – 1
∴ 2 X = 11 Y – 1 …………. (ii)
Step 9: By comparing twice of equation (i) with equation (ii), we get:
2 (Y 2 + 2) = 11 Y – 1
∴ 2 Y 2 + 4 = 11 Y – 1
∴ 2 Y 2 – 11 Y + 5 = 0
∴ 2 Y 2 – 10 Y – Y + 5 = 0
∴ 2 Y (Y – 5) – (Y – 5) = 0
∴ (Y – 5) (2 Y – 1) = 0
∴ Y = 5 and 1/2 years
Since, age can not be in fraction value, hence we reject Y = 1/2 and accept Y = 5
∴ Y = 5 years (Nisha’s age)
Step 10: By substituting value of Y in equation (ii), we get:
∵ 2 X = 11 Y – 1
∴ 2 X = 11 (5) – 1 = 54
∴ X = 27 years
Step 11: Sum of their present ages = Asha’s present age + Nisha’s present age
= 27 + 5 = 32 years
Therefore, the sum of their present ages is 32 years.
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Check: After solving, let’s quickly put our answers and cross-check with the given conditions:
Our answer is Asha’s present age = 27 and Nisha’s present age = 2
First condition: “Asha’s age is 2 more than the square of her daughter Nisha’s age” i.e. 27 = 5 2 + 2 …. satisfied
Second condition: “When Nisha grows to her mother’s present age” i.e. (27 – 5 = 22 years later), Asha will be 49 and Nisha will be 27 (of Asha’s age). Then, “Asha’s age would be one year less than 10 times the present age of Nisha’s age”: 49 = 10 x 5 – 1 … satisfied.
Therefore, our answers are correct.