(a) Estimate the value of $$ \frac{49 \cdot 27+11 \cdot 15}{15 \cdot 24-3 \cdot 06} $$ by rounding each number to the nearest whole number. (b) Calculate the error in your estimated answer, correct to four decimal places. c) Calculate the percentage error in your estimated answer, correct to one decimal place.

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UpStudy AI · Accepted Answer

Let's tackle each part of the problem step-by-step.

### **Part (a): Estimate the Value**

We need to estimate the value of the expression:

$$
\frac{49 \times 27 + 11 \times 15}{15 \times 24 - 3 \times 6}
$$

**Step 1: Round Each Number to the Nearest Ten**

For estimation purposes, it's common to round each number to the nearest ten to simplify calculations.

- **Numerator:**
  - $$49 $$ rounds to $$50$$
  - $$27 $$ rounds to $$30$$
  - $$11 $$ rounds to $$10$$
  - $$15 $$ rounds to $$20$$

- **Denominator:**
  - $$15 $$ rounds to $$20$$
  - $$24 $$ rounds to $$20$$
  - $$3 $$ rounds to $$0$$ (since it's closer to $$0$$)
  - $$6 $$ rounds to $$0$$

**Step 2: Substitute Rounded Values into the Expression**

$$
\frac{50 \times 30 + 10 \times 20}{20 \times 20 - 0 \times 0} = \frac{1500 + 200}{400 - 0} = \frac{1700}{400}
$$

**Step 3: Simplify**

$$
\frac{1700}{400} = 4.25
$$

**Estimated Value:** $$ \boxed{4.25} $$

---

### **Part (b): Calculate the Error**

First, calculate the **exact value** of the original expression.

**Step 1: Compute the Exact Numerator and Denominator**

- **Numerator:**
  $$
  49 \times 27 + 11 \times 15 = 1323 + 165 = 1488
  $$

- **Denominator:**
  $$
  15 \times 24 - 3 \times 6 = 360 - 18 = 342
  $$

**Step 2: Compute the Exact Value**

$$
\frac{1488}{342} \approx 4.3509
$$

**Step 3: Calculate the Error**

$$
\text{Error} = |\text{Estimated Value} - \text{Exact Value}| = |4.25 - 4.3509| = 0.1009
$$

**Error:** $$ \boxed{0.1009} $$

---

### **Part (c): Calculate the Percentage Error**

$$
\text{Percentage Error} = \left( \frac{\text{Error}}{\text{Exact Value}} \right) \times 100 = \left( \frac{0.1009}{4.3509} \right) \times 100 \approx 2.3\%
$$

**Percentage Error:** $$ \boxed{2.3\%} $$
**Part (a):** The estimated value is 4.25. **Part (b):** The error is 0.1009. **Part (c):** The percentage error is 2.3%.

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