Let’s solve each question step by step.

Question 1

Walter currently runs the 40-yard dash in $7.2$ seconds. He believes he can reduce his time by $0.2$ seconds per week. We are tasked to find an expression for his time after $w$ weeks.

Step 1: Write the equation

Each week, Walter’s time decreases by $0.2w$. After $w$ weeks, his time will be: $\text{Time} = 7.2 - 0.2w$

Answer: D. $7.2 - 0.2w$

Question 2

A piece of yarn 4 yards long is cut in half, and each half is cut into fourths. We need to determine the length of each piece in inches.

Step 1: Convert 4 yards to inches

Since $1 \, \text{yard} = 36 \, \text{inches}$, $4 \, \text{yards}$ equals: $4 \times 36 = 144 \, \text{inches}.$

Step 2: Cut into halves

Cutting 144 inches in half gives: $\frac{144}{2} = 72 \, \text{inches}.$

Step 3: Cut each half into fourths

Each half of 72 inches is divided into 4 parts: $\frac{72}{4} = 18 \, \text{inches}.$

Answer: B. $18 \, \text{inches}$

Question 3

A ride-sharing service charges a base fee of $\$2.40$ per ride and $\$0.30$ per mile. Edward paid $\$3.60$ for one ride. We need to calculate the number of miles Edward rode.

Step 1: Write the cost equation

The total cost is: $\text{Cost} = \text{Base fee} + (\text{Cost per mile} \times \text{Miles})$ Substitute the known values: $3.60 = 2.40 + 0.30 \times \text{Miles}.$

Step 2: Solve for miles

Subtract $2.40$ from both sides: $3.60 - 2.40 = 0.30 \times \text{Miles}.$ $1.20 = 0.30 \times \text{Miles}.$ Divide by $0.30$: $\text{Miles} = \frac{1.20}{0.30} = 4.$

Answer: C. $4 \, \text{miles}$

Question 4

Tiki cycled 13 fewer miles than Irina, and together they cycled 51 miles. We need to find how many miles Irina cycled.

Step 1: Let Irina’s miles be $x$

Tiki cycled $x - 13$ miles. Together, they cycled 51 miles: $x + (x - 13) = 51.$

Step 2: Solve for $x$

Simplify: $2x - 13 = 51.$ Add 13 to both sides: $2x = 64.$ Divide by 2: $x = 32.$

Irina cycled 32 miles.

Answer: B. $32 \, \text{miles}$

Would you like further clarification or detailed explanations for any step? Here are related concepts:

1. How can we generalize the reduction in time in Question 1 for non-linear improvements?
2. How do unit conversions (yards to inches) apply in larger contexts?
3. How could Question 3 change if Edward had a discount on the base fee?
4. How can systems of equations be used to solve similar problems like Question 4?

Tip: Always double-check unit conversions in measurement problems—they’re a common source of errors!