ether? he sum of the ages of Mr. Daniel and his laughter is 60 years. 5 years ago, the ratio of heir ages was $$ 1: 4 $$. a) How old are they now?

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UpStudy AI · Accepted Answer

It appears there might have been a slight typo in your question. Assuming you meant:

**"The sum of the ages of Mr. Daniel and his [child] is 60 years. Five years ago, the ratio of their ages was $$1:4$$.**

Let's solve for their current ages.

### Let’s Define the Variables:
- Let $$ x $$ be Mr. Daniel's current age.
- Let $$ y $$ be his child's current age.

### Given:
1. **Sum of their current ages:**
   $$
   x + y = 60 \quad \text{(Equation 1)}
   $$
2. **Five years ago, the ratio of their ages was $$1:4$$:**
   $$
   \frac{x - 5}{y - 5} = \frac{1}{4} \quad \text{(Equation 2)}
   $$

### Solving the Equations:

**From Equation 2:**
$$
4(x - 5) = y - 5
$$
$$
4x - 20 = y - 5
$$
$$
4x - y = 15 \quad \text{(Equation 3)}
$$

**Now, substitute Equation 1 into Equation 3:**
$$
4x - (60 - x) = 15
$$
$$
4x - 60 + x = 15
$$
$$
5x - 60 = 15
$$
$$
5x = 75
$$
$$
x = 15
$$

**This would imply:**
$$
y = 60 - 15 = 45
$$

**However, this result suggests that Mr. Daniel is 15 years old and his child is 45, which isn't logical.**

**Re-evaluating the Ratio:**
It's more plausible that **Mr. Daniel is the older individual**. Therefore, the correct interpretation of the ratio should be:
$$
\frac{y - 5}{x - 5} = \frac{1}{4}
$$
$$
y - 5 = \frac{1}{4}(x - 5)
$$
$$
4(y - 5) = x - 5
$$
$$
4y - 20 = x - 5
$$
$$
4y - x = 15 \quad \text{(Equation 4)}
$$

**Substituting Equation 1 into Equation 4:**
$$
4y - (60 - y) = 15
$$
$$
4y - 60 + y = 15
$$
$$
5y = 75
$$
$$
y = 15
$$
$$
x = 60 - 15 = 45
$$

### **Solution:**
- **Mr. Daniel's Current Age:** 45 years
- **His Child's Current Age:** 15 years

### **Verification:**
- **Five years ago:**
  - Mr. Daniel was $$45 - 5 = 40$$ years old.
  - His child was $$15 - 5 = 10$$ years old.
- **Ratio of their ages five years ago:** $$40:10 = 4:1$$, which aligns with the given ratio when interpreted correctly.

**Therefore, Mr. Daniel is currently 45 years old, and his child is 15 years old.**
Mr. Daniel is 45 years old, and his child is 15 years old.

ether? he sum of the ages of Mr. Daniel and his laughter is 60 years. 5 years ago, the ratio of heir ages was \( 1: 4 \). a) How old are they now?

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