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Amita is 2 years older than her friend Amrita. Amita's father is twice as old as Amita, and Amrita is twice as old as her sister. The ages of Amita's father and Amrita's sister differ by 43 years. The sum of the ages (in years) of Amita and Amrita is:
54
50
64
68
54
Represent all ages in terms of one variable (e.g., Amrita = x) to form a simple linear equation: 2(x + 2) - x/2 = 43.
Represent all ages in terms of one variable (e.g., Amrita = x) to form a simple linear equation: 2(x + 2) - x/2 = 43.
Define Variables
Let Amrita's age be x. Since Amita is 2 years older, Amita = x + 2. Amrita's sister is half of Amrita's age, so Sister = x/2. Amita's father is twice Amita's age, so Father = 2(x + 2) = 2x + 4.
Formulate Equation
The difference between Father's age and Sister's age is 43. Thus, (2x + 4) - x/2 = 43. Multiplying by 2 to clear the fraction: 4x + 8 - x = 86.
Solve for x
3x + 8 = 86 тЖТ 3x = 78 тЖТ x = 26. Amrita is 26 and Amita is 26 + 2 = 28.
Calculate Final Sum
Sum of Amita and Amrita ages = 28 + 26 = 54.
B: 50 is incorrect as it implies x=24, leading to a difference of 39 instead of 43. C: 64 is incorrect as it implies x=31, leading to a difference of 52.5 instead of 43. D: 68 is incorrect as it implies x=33, leading to a difference of 56.5 instead of 43.
A is correct because solving the linear equation (2x + 4) - 0.5x = 43 yields Amrita=26 and Amita=28, which sum to 54.
In age problems involving fractions, always assign the variable to the person who makes the other ages expressible as whole numbers to avoid complex arithmetic.
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