Q is as much younger than r as he is older than t. if the sum of the ages of r and t is 50 years, what is definitely the difference between r and q's age?

General
1

q is as much younger than r as he is older than t. if the sum of the ages of r and t is 50 years, what is definitely the difference between r and q’s age?

2

LectureNotes said Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q’s age?

Answer:
To solve this problem, let’s define the ages of R, Q, and T using algebraic expressions.

  1. Define the Variables:

    • Let ( R ) be the age of R.
    • Let ( Q ) be the age of Q.
    • Let ( T ) be the age of T.

  2. Set up the Equations:

    • According to the problem, Q is as much younger than R as he is older than T. This can be expressed as:
      R - Q = Q - T
    • The sum of the ages of R and T is given as 50 years:
      R + T = 50

  3. Solve for Q:

    • From the first equation ( R - Q = Q - T ), we can rearrange it to:
      R - Q = Q - T \implies R + T = 2Q

  4. Substitute and Solve:

    • We already have ( R + T = 50 ). Substitute this into the equation ( R + T = 2Q ):
      50 = 2Q \implies Q = 25

  5. Find the Difference Between R and Q’s Age:

    • From the equation ( R + T = 50 ) and ( R + T = 2Q ), we know ( 2Q = 50 ) so ( Q = 25 ).
    • Substitute ( Q = 25 ) back into ( R - Q = Q - T ):
      R - 25 = 25 - T
    • Since ( R + T = 50 ), we can substitute T = 50 - R into R - 25 = 25 - T :
      R - 25 = 25 - (50 - R) \implies R - 25 = 25 - 50 + R \implies R - 25 = -25 + R
    • This simplifies to:
      R - 25 = R - 25

Since the equation holds true, we can conclude that the difference between R and Q’s age is:

\boxed{25}

Therefore, the definite difference between R and Q’s age is 25 years.