is younger than Tom, but older than Harry. Their ages in years are consecutive odd integers. 's mother is three times as old as , and she is also 13 years older than twice as old as Tom. How old are Harry, , and Tom

H < D < T

H + 2 = D ... D + 2 = T

3 D = 2 T + 13

3 D = 2 (D + 2) + 13 = 2 D + 17

D = 's age

T = Tom's age

H = Harry's age

M = Mother's age

D < T

D > H

D = T - 2 Add 2 to both sides

D + 2 = T - 2 + 2

D + 2 = T

T = D + 2

M = 3 D

also:

M = 2 T + 13 = 2 ( D + 2 ) + 13 = 2 * D + 2 * 2 + 13 = 2 D + 4 + 13 = 2 D + 17

M = M

3 D = 2 D + 17 Subtract 2 D to both sides

3 D - 2 D = 2 D + 17 - 2 D

D = 17 yrs

T = D + 2 = 17 + 2 = 19 yrs

D = H + 2 Subtract 2 to both sides

D - 2 = H + 2 - 2

D - 2 = H

H = D - 2

H = 17 - 2 = 15 yrs

Proof:

's mother is three times as old as :

M = 3 D = 3 * 17 = 51 yrs

's mother is 13 years older than twice as old as Tom:

M = 2 T + 13 = 2 * 19 + 13 = 38 + 13 = 51 yrs

To solve this problem, let's break it down step by step.

Step 1: Define the variables
Let's assign variables to the ages of Harry, , and Tom. We'll let Harry's age be 'h', 's age be 'd', and Tom's age be 't'.

Step 2: Set up equations using the given information
According to the given information, is younger than Tom, but older than Harry, and their ages are consecutive odd integers. This can be translated into the following equations:

1. is younger than Tom: d < t
2. is older than Harry: d > h
3. Consecutive odd integers: t = d + 2 and h = d - 2

Step 3: Use the information about 's mother to set up another equation
According to the problem, 's mother is three times as old as , and she is also 13 years older than twice as old as Tom. Let's translate this into an equation:

4. 's mother's age: m = 3d
5. 's mother is 13 years older than twice as old as Tom: m = 2t + 13

Step 4: Solve the system of equations
Now we have a system of equations consisting of equations (1) to (5). We can solve this system to find the ages of Harry, , and Tom.

Substituting the values for t and h from equations (3) into equations (1) and (2), we get:
d < d + 2 and d > d - 2, which simplify to:
0 < 2 and 0 > -2

Since both inequalities are true, we know that 's age can be any positive odd number greater than 0.

Now, substituting the value of t from equation (3) into equation (5), we get:
m = 2(d + 2) + 13
m = 2d + 4 + 13
m = 2d + 17

Since 's mother's age is a multiple of 3, we know that m must be a multiple of 3. The smallest value of d that satisfies this condition is 2 (since 2 * 3 = 6).

Therefore, 's age (d) is 2, Tom's age (t) is 4, and Harry's age (h) is 0.

So, Harry is 0 years old, is 2 years old, and Tom is 4 years old.