The ratio of the ages of Marty and her mother is 1: 4 respectively. While the same for her sister Missy and their mother one year from now will be 1: 3 respectively. Seven years from now, the ratio of the sum of the ages of the two sisters and the age of their mother will be 13: 17 respectively. How old was their mother when Missy was born?

Pick ONE option

A) 31 years

B) 28 years

C) 30 years

D) 33 years

Let the current ages of Marty, Missy, and their mother be represented by:

- Marty's age: \( M \)
- Missy's age: \( S \)
- Mother's age: \( X \)

### Step 1: The ratio of Marty's and her mother's ages

From the problem, the current ratio of Marty's age to her mother's age is \( 1:4 \). Thus,

\[
M = \frac{1}{4}X \quad \text{or equivalently,} \quad 4M = X.
\]

### Step 2: The ratio of Missy's and her mother's ages one year from now

The ratio of Missy's age to her mother's age one year from now will be \( 1:3 \). Therefore, we have the equation:

\[
\frac{S + 1}{X + 1} = \frac{1}{3} \quad \text{which simplifies to} \quad 3(S + 1) = X + 1.
\]
Expanding this:

\[
3S + 3 = X + 1 \quad \text{or equivalently,} \quad 3S - X = -2.
\]

### Step 3: The ratio of the sum of the sisters' ages to their mother's age seven years from now

Seven years from now, the sum of Marty’s and Missy’s ages compared to their mother’s age will be \( 13:17 \). Therefore, the equation is:

\[
\frac{(M + 7) + (S + 7)}{X + 7} = \frac{13}{17}.
\]
Simplifying this:

\[
\frac{M + S + 14}{X + 7} = \frac{13}{17},
\]
which gives:

\[
17(M + S + 14) = 13(X + 7).
\]
Expanding both sides:

\[
17(M + S + 14) = 17M + 17S + 238
\]
and

\[
13(X + 7) = 13X + 91.
\]
Thus, the equation becomes:

\[
17M + 17S + 238 = 13X + 91,
\]
which simplifies to:

\[
17M + 17S + 147 = 13X.
\]

### Step 4: Solving the system of equations

Now, we have the following system of equations:
1. \( 4M = X \)
2. \( 3S - X = -2 \)
3. \( 17M + 17S + 147 = 13X \)

Substituting \( X = 4M \) from equation (1) into equation (2):

\[
3S - 4M = -2 \quad \text{or equivalently,} \quad 3S = 4M - 2 \quad \text{or} \quad S = \frac{4M - 2}{3}.
\]

Substitute \( S = \frac{4M - 2}{3} \) and \( X = 4M \) into equation (3):

\[
17M + 17\left(\frac{4M - 2}{3}\right) + 147 = 13(4M).
\]
Simplifying this:

\[
17M + \frac{68M - 34}{3} + 147 = 52M.
\]
Multiplying through by 3 to eliminate the fraction:

\[
51M + 68M - 34 + 441 = 156M.
\]
Simplifying further:

\[
119M + 407 = 156M,
\]
which gives:

\[
407 = 37M.
\]
Thus,

\[
M = \frac{407}{37} = 11.
\]

### Step 5: Finding the mother's age and Missy's age

Using \( M = 11 \) in \( X = 4M \), we find:

\[
X = 4 \times 11 = 44.
\]

Now, using \( M = 11 \) in \( S = \frac{4M - 2}{3} \), we find:

\[
S = \frac{4(11) - 2}{3} = \frac{44 - 2}{3} = \frac{42}{3} = 14.
\]

### Step 6: Finding the mother's age when Missy was born

Missy's current age is 14, so her mother was \( 44 - 14 = 30 \) years old when Missy was born.

Thus, the correct answer is:

**C) 30 years**

---

Would you like further details on any part of the solution?

Here are 5 related questions to expand on this topic:
1. How would the solution change if the ratio of Marty's and her mother's ages was 1:3?
2. Can you generalize the approach for solving age-related problems with multiple unknowns?
3. What is the mathematical reasoning behind ratios and proportions in age problems?
4. How would the solution be different if Missy was younger than Marty?
5. Could you solve this problem by trial and error instead of algebraically?

**Tip:** For age-related problems, always set up relationships between variables before jumping into calculations. This helps in organizing the information effectively!

The ratio of the ages of Marty and her mother is 1:4 respectively. While the same for her sister Missy and their mother one year from now will be 1:3 respectively. Seven years from now, the ratio of the sum of the ages of the two sisters and the age of their mother will be 13:17 respectively. How old was their mother when Missy was born? A) 31 years B) 28 years C) 30 years D) 33 years

Learn how to solve a complex age problem involving ratios and systems of equations. The question explores the relationships between the ages of Marty, Missy, and their mother, requiring algebraic reasoning to determine the mother's age when Missy was born. Ideal for students in Grades 9-12.

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