49. Age differences and ratios problems - step by step solution procedure and detailed explanation for each problem

49. Age differences and ratios problems - step by step solution procedure and detailed explanation for each problem

Age differences and ratios problems involve relationships between the ages of two or more individuals, either at the present time or at different points in time (past or future). These problems typically use differences in ages, ratios of ages, or changes in ages over time to set up equations and solve for unknown ages. They often require careful handling of time shifts and consistent variable definitions.

Below are 10 age differences and ratios problems with step-by-step solutions and detailed explanations. Each problem includes the problem statement, solution procedure, verification, and an explanation of the concept. The problems are designed to be diverse, covering various scenarios involving differences, ratios, past and future ages, and multiple individuals. They are distinct from previous problem types and avoid any unrelated contexts (e.g., alcohol or mixing solutions).

Problem 1

The ratio of Alice’s age to Bob’s age is 3:5. If the difference between their ages is 8 years, how old are they?

Step-by-Step Solution:

Define variables: Let Alice’s age be ( 3x ) and Bob’s age be ( 5x ), based on the ratio 3:5.
Set up the equation: The difference in their ages is 8 years.
5x - 3x = 8
2x = 8 \implies x = 4
Calculate ages:
- Alice’s age:
  3x = 3 \cdot 4 = 12
- Bob’s age:
  5x = 5 \cdot 4 = 20
Verify: Ratio =
\frac{12}{20} = \frac{3}{5}
. Difference =
20 - 12 = 8
. Both conditions are satisfied.
Answer: Alice is 12 years old, and Bob is 20 years old.

Explanation: The ratio provides a proportional relationship, and the age difference gives a concrete value to solve for the scaling factor ( x ). The equation balances the difference while maintaining the ratio, a common approach in ratio-based age problems.

Problem 2

Five years ago, the age difference between a father and his son was 30 years. Now, the ratio of their ages is 7:2. How old are they now?

Step-by-Step Solution:

Define variables: Let the father’s current age be ( f ) and the son’s current age be ( s ).
Set up equations:
- Five years ago, their age difference was 30:
  (f - 5) - (s - 5) = 30 \implies f - s = 30
- Current age ratio:
  \frac{f}{s} = \frac{7}{2} \implies f = \frac{7}{2}s
Solve the system:
- Substitute
  f = \frac{7}{2}s
  into
  f - s = 30
  :
  \frac{7}{2}s - s = 30
  \frac{7s - 2s}{2} = 30 \implies \frac{5s}{2} = 30 \implies 5s = 60 \implies s = 12
- Find ( f ):
  f = \frac{7}{2} \cdot 12 = 42
Verify: Current ages: father = 42, son = 12. Ratio =
\frac{42}{12} = \frac{7}{2}
. Five years ago: father =
42 - 5 = 37
, son =
12 - 5 = 7
. Difference =
37 - 7 = 30
.
Answer: The father is 42 years old, and the son is 12 years old.

Explanation: The problem combines a past age difference with a current ratio. The difference remains constant over time (

f - s = 30

), and the ratio applies to the present, allowing us to solve for current ages. Time shifts are critical in such problems.

Problem 3

The difference between Clara’s and David’s ages is 10 years. In 5 years, the ratio of their ages will be 5:3. How old are they now?

Step-by-Step Solution:

Define variables: Let Clara’s current age be ( c ) and David’s current age be ( d ).
Set up equations:
- Age difference:
  c - d = 10
  (assuming Clara is older; if not, test
  d - c = 10
  ).
- In 5 years, ratio:
  \frac{c + 5}{d + 5} = \frac{5}{3}
Solve the system:
- From the first equation:
  c = d + 10
- Substitute into the ratio equation:
  \frac{(d + 10) + 5}{d + 5} = \frac{5}{3}
  \frac{d + 15}{d + 5} = \frac{5}{3}
  3 (d + 15) = 5 (d + 5)
  3d + 45 = 5d + 25
  45 - 25 = 5d - 3d \implies 20 = 2d \implies d = 10
- Find ( c ):
  c = d + 10 = 10 + 10 = 20
Verify: Current ages: Clara = 20, David = 10. Difference =
20 - 10 = 10
. In 5 years: Clara =
20 + 5 = 25
, David =
10 + 5 = 15
. Ratio =
\frac{25}{15} = \frac{5}{3}
.
Answer: Clara is 20 years old, and David is 10 years old.

Explanation: The age difference is constant, and the future ratio provides a relationship to solve for current ages. Assuming Clara is older simplifies the setup, but testing the reverse (David older) would yield no solution since ages must be positive and consistent with the ratio.

Problem 4

Ten years ago, the ratio of Emma’s age to Frank’s age was 1:2. Now, their age difference is 15 years. How old are they now?

Step-by-Step Solution:

Define variables: Let Emma’s current age be ( e ) and Frank’s current age be ( f ).
Set up equations:
- Ten years ago, ratio:
  \frac{e - 10}{f - 10} = \frac{1}{2} \implies 2 (e - 10) = f - 10 \implies 2e - 20 = f - 10 \implies f = 2e - 10
- Current age difference:
  f - e = 15
  (assuming Frank is older).
Solve the system:
- Substitute
  f = 2e - 10
  into
  f - e = 15
  :
  (2e - 10) - e = 15
  e - 10 = 15 \implies e = 25
- Find ( f ):
  f = 2 \cdot 25 - 10 = 50 - 10 = 40
Verify: Current ages: Emma = 25, Frank = 40. Difference =
40 - 25 = 15
. Ten years ago: Emma =
25 - 10 = 15
, Frank =
40 - 10 = 30
. Ratio =
\frac{15}{30} = \frac{1}{2}
.
Answer: Emma is 25 years old, and Frank is 40 years old.

Explanation: The past ratio and current difference provide two equations. The ratio from 10 years ago reflects their ages at that time, and the constant age difference allows us to solve for current ages. The solution confirms consistency across time.

Problem 5

The current ages of two sisters are in the ratio 4:5. In 8 years, their ages will be in the ratio 6:7. How old are they now?

Step-by-Step Solution:

Define variables: Let the younger sister’s current age be ( 4x ) and the older sister’s be ( 5x ), based on the ratio 4:5.
Set up the equation: In 8 years, the ratio is 6:7.
\frac{4x + 8}{5x + 8} = \frac{6}{7}
Solve the equation:
7 (4x + 8) = 6 (5x + 8)
28x + 56 = 30x + 48
56 - 48 = 30x - 28x
8 = 2x \implies x = 4
- Younger sister’s age:
  4x = 4 \cdot 4 = 16
- Older sister’s age:
  5x = 5 \cdot 4 = 20
Verify: Current ages: 16 and 20. Ratio =
\frac{16}{20} = \frac{4}{5}
. In 8 years: 16 + 8 = 24, 20 + 8 = 28. Ratio =
\frac{24}{28} = \frac{6}{7}
.
Answer: The younger sister is 16 years old, and the older sister is 20 years old.

Explanation: The current and future ratios provide relationships between the sisters’ ages. The equation accounts for the age increase over 8 years, solving for the scaling factor ( x ). The solution ensures both ratios are satisfied.

Problem 6

The difference between a mother’s and daughter’s ages is 24 years. Six years ago, the ratio of their ages was 5:1. How old are they now?

Step-by-Step Solution:

Define variables: Let the mother’s current age be ( m ) and the daughter’s current age be ( d ).
Set up equations:
- Age difference:
  m - d = 24
- Six years ago, ratio:
  \frac{m - 6}{d - 6} = \frac{5}{1} \implies m - 6 = 5 (d - 6)
Solve the system:
- From the first equation:
  m = d + 24
- Substitute into the second equation:
  (d + 24) - 6 = 5 (d - 6)
  d + 18 = 5d - 30
  18 + 30 = 5d - d
  48 = 4d \implies d = 12
- Find ( m ):
  m = d + 24 = 12 + 24 = 36
Verify: Current ages: mother = 36, daughter = 12. Difference =
36 - 12 = 24
. Six years ago: mother =
36 - 6 = 30
, daughter =
12 - 6 = 6
. Ratio =
\frac{30}{6} = \frac{5}{1}
.
Answer: The mother is 36 years old, and the daughter is 12 years old.

Explanation: The constant age difference and past ratio allow us to set up a system of equations. The ratio from six years ago reflects their ages at that time, and solving ensures consistency with the current difference.

Problem 7

The ratio of Tom’s age to Jerry’s age is 2:3. In 10 years, their ages will differ by 5 years. How old are they now?

Step-by-Step Solution:

Define variables: Let Tom’s current age be ( 2x ) and Jerry’s current age be ( 3x ), based on the ratio 2:3.
Set up the equation: In 10 years, their age difference is 5 years.
- In 10 years: Tom’s age =
  2x + 10
  , Jerry’s age =
  3x + 10
- Difference:
  (3x + 10) - (2x + 10) = 5
  3x + 10 - 2x - 10 = 5
  x = 5
Calculate ages:
- Tom’s age:
  2x = 2 \cdot 5 = 10
- Jerry’s age:
  3x = 3 \cdot 5 = 15
Verify: Current ages: Tom = 10, Jerry = 15. Ratio =
\frac{10}{15} = \frac{2}{3}
. In 10 years: Tom =
10 + 10 = 20
, Jerry =
15 + 10 = 25
. Difference =
25 - 20 = 5
.
Answer: Tom is 10 years old, and Jerry is 15 years old.

Explanation: The current ratio and future difference provide the necessary relationships. The age difference in the future simplifies to the current difference (since

(3x + 10) - (2x + 10) = x

), making the equation straightforward.

Problem 8

Eight years ago, the age difference between a grandfather and his grandson was 50 years. Now, the ratio of their ages is 13:3. How old are they now?

Step-by-Step Solution:

Define variables: Let the grandfather’s current age be ( g ) and the grandson’s current age be ( s ).
Set up equations:
- Eight years ago, difference:
  (g - 8) - (s - 8) = 50 \implies g - s = 50
- Current ratio:
  \frac{g}{s} = \frac{13}{3} \implies g = \frac{13}{3}s
Solve the system:
- Substitute
  g = \frac{13}{3}s
  into
  g - s = 50
  :
  \frac{13}{3}s - s = 50
  \frac{13s - 3s}{3} = 50 \implies \frac{10s}{3} = 50 \implies 10s = 150 \implies s = 15
- Find ( g ):
  g = \frac{13}{3} \cdot 15 = 65
Verify: Current ages: grandfather = 65, grandson = 15. Ratio =
\frac{65}{15} = \frac{13}{3}
. Eight years ago: grandfather =
65 - 8 = 57
, grandson =
15 - 8 = 7
. Difference =
57 - 7 = 50
.
Answer: The grandfather is 65 years old, and the grandson is 15 years old.

Explanation: The past difference and current ratio form a system of equations. The difference remains constant over time, and the ratio applies to the present, allowing us to solve for current ages with a clear relationship.

Problem 9

The current ages of two friends are in the ratio 5:6. Twelve years ago, their ages were in the ratio 3:4. How old are they now?

Step-by-Step Solution:

Define variables: Let the first friend’s current age be ( 5x ) and the second friend’s be ( 6x ), based on the ratio 5:6.
Set up the equation: Twelve years ago, the ratio was 3:4.
\frac{5x - 12}{6x - 12} = \frac{3}{4}
Solve the equation:
4 (5x - 12) = 3 (6x - 12)
20x - 48 = 18x - 36
20x - 18x = 48 - 36
2x = 12 \implies x = 6
- First friend’s age:
  5x = 5 \cdot 6 = 30
- Second friend’s age:
  6x = 6 \cdot 6 = 36
Verify: Current ages: 30 and 36. Ratio =
\frac{30}{36} = \frac{5}{6}
. Twelve years ago: 30 - 12 = 18, 36 - 12 = 24. Ratio =
\frac{18}{24} = \frac{3}{4}
.
Answer: The first friend is 30 years old, and the second friend is 36 years old.

Explanation: The current and past ratios provide two relationships across time. The equation accounts for the age decrease 12 years ago, solving for the scaling factor ( x ). The solution ensures both ratios are consistent.

Problem 10

The difference between a teacher’s and a student’s ages is 20 years. In 4 years, the ratio of their ages will be 7:3. How old are they now?

Step-by-Step Solution:

Define variables: Let the teacher’s current age be ( t ) and the student’s current age be ( s ).
Set up equations:
- Age difference:
  t - s = 20
- In 4 years, ratio:
  \frac{t + 4}{s + 4} = \frac{7}{3}
Solve the system:
- From the first equation:
  t = s + 20
- Substitute into the ratio equation:
  \frac{(s + 20) + 4}{s + 4} = \frac{7}{3}
  \frac{s + 24}{s + 4} = \frac{7}{3}
  3 (s + 24) = 7 (s + 4)
  3s + 72 = 7s + 28
  72 - 28 = 7s - 3s \implies 44 = 4s \implies s = 11
- Find ( t ):
  t = s + 20 = 11 + 20 = 31
Verify: Current ages: teacher = 31, student = 11. Difference =
31 - 11 = 20
. In 4 years: teacher =
31 + 4 = 35
, student =
11 + 4 = 15
. Ratio =
\frac{35}{15} = \frac{7}{3}
.
Answer: The teacher is 31 years old, and the student is 11 years old.

Explanation: The constant age difference and future ratio provide a system to solve for current ages. The future ratio accounts for the age increase, and the solution ensures consistency with the difference.

General Notes on Age Differences and Ratios Problems

Key Concept: These problems involve relationships between ages, using differences (which remain constant over time) and ratios (which change as ages increase). They often require setting up equations based on current, past, or future conditions.
Key Steps:
1. Define variables for the current ages of individuals.
2. Set up equations based on given differences or ratios, accounting for time shifts (e.g., past or future ages).
3. Solve the system of equations, typically using substitution or elimination.
4. Verify by checking all conditions (e.g., ratios, differences) at the specified times.
Common Equations:
- Difference:
  a - b = d
  (remains constant over time).
- Ratio:
  \frac{a}{b} = \frac{m}{n} \implies a = \frac{m}{n}b
  .
- Time shift: For ( t ) years ago, ages are
  a - t
  ,
  b - t
  ; for ( t ) years later, ages are
  a + t
  ,
  b + t
  .
Challenges:
- Ensure the direction of the difference (e.g., who is older) is clear. If ambiguous, test both cases.
- Handle time shifts carefully to avoid errors in past or future ages.
- Verify solutions to catch inconsistencies, as ages must be positive and realistic.
Alternative Approach: For ratio problems, alligation can sometimes be applied to find ratios of ages at different times, but equations are typically more straightforward.
Applications: These problems model real-world scenarios like family relationships, educational settings, or generational gaps, requiring logical reasoning about age progression.

These problems cover a range of scenarios, from simple ratio-difference combinations to complex past-future relationships, demonstrating the versatility of age problems. If you need more problems, variations, or further clarification, let me know!

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49. Age differences and ratios problems - step by step solution procedure and detailed explanation for each problem

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