A runner's racecourse has a total distance of 10 mi. Race officials want to mark the course every 275 yd with a flag, including one at the finish line. (The starting line does not need a flag.)

How many flags are needed?

First, fill in the blanks on the left side of the equation using three of the ratios shown. Then write your answer on the right side of the equation.



Answer :

Sure, let's go through the steps in detail to solve this problem.

Step 1: Convert the race distance from miles to yards.
- 1 mile = 1760 yards
- Race distance in miles = 10 miles
- To convert to yards: 10 miles × 1760 yards/mile

So, the total race distance in yards is:
[tex]\[ 10 \times 1760 = 17600 \text{ yards} \][/tex]

Step 2: Determine the intervals at which the flags need to be placed.
- Flag distance: 275 yards

Step 3: Calculate the number of intervals (flags) needed to cover the entire race distance, not forgetting to place a flag at the finish line.

[tex]\[ \text{Number of flags} = \left\lfloor \frac{\text{Total race distance in yards}}{\text{Flag distance in yards}} \right\rfloor + 1 \][/tex]

Where:
- Total race distance in yards = 17600 yards
- Flag distance in yards = 275 yards

Substituting these values in, we get:

[tex]\[ \left\lfloor \frac{17600}{275} \right\rfloor + 1 \][/tex]

[tex]\[ = 64 + 1 \][/tex]

[tex]\[ = 65 \][/tex]

So, the number of flags needed is:
[tex]\[ 65 \][/tex]

To summarize, the racecourse is 17600 yards long, and the officials need to place 65 flags every 275 yards, including one at the finish line.