# 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?

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?

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?

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.
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