Sarah is 16years older than Sam. in10 years time, the ratio of Sam's age to Sarah age will be 2:3respectively. how old is Sarah new

Mathematics

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Answer

38 years38\ \text{years}

Step-by-step solution

6 steps

  1. Define variables for current ages

    Let Sam's current age be SS years.
    Let Sarah's current age be AA years.

  2. Formulate the first equation based on the age difference

    "Sarah is 16 years older than Sam."

    A=S+16(1)A = S + 16 \quad (1)

  3. Formulate the second equation based on the future ratio

    "In 10 years time, the ratio of Sam's age to Sarah age will be 2:3 respectively."
    In 10 years, Sam's age will be S+10S + 10.
    In 10 years, Sarah's age will be A+10A + 10.
    The ratio is:

    S+10A+10=23(2)\frac{S + 10}{A + 10} = \frac{2}{3} \quad (2)

  4. Substitute the first equation into the second equation

    Substitute A=S+16A = S + 16 from (1) into (2):

    S+10(S+16)+10=23\frac{S + 10}{(S + 16) + 10} = \frac{2}{3} S+10S+26=23\frac{S + 10}{S + 26} = \frac{2}{3}

  5. Solve for Sam's current age (S)

    Cross-multiply:

    3(S+10)=2(S+26)3(S + 10) = 2(S + 26) 3S+30=2S+523S + 30 = 2S + 52

    Subtract 2S2S from both sides:

    S+30=52S + 30 = 52

    Subtract 30 from both sides:

    S=22S = 22

    Sam's current age is 22 years.

  6. Calculate Sarah's current age (A)

    Using equation (1), A=S+16A = S + 16:

    A=22+16A = 22 + 16 A=38A = 38

    Sarah's current age is:

    38 years38\ \text{years}
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