Question

Math

Posted 5 months ago

```
Rohit is recapping his track season where he ran in a total of 24 races.
His coach tells him that 4 of his races were 100 meters in length and 2 races were 400 meters in length. He remembers that during the last 12 races of the season, he was expected to run a total of 4200 meters. He's also willing to assume that the lengths of these 12 races are representative of the rest of the races.
He isn't sure how many of the other races were 200 meters long and how many were 800 meters long.
Find the 2 missing numbers from the data table below.
\begin{tabular}{cc}
Race length (meters) & Races \\
\hline 100 & 4 \\
200 & \\
400 & 2 \\
800 & \\
Total & 24
\end{tabular}
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Combine the known quantities of races and the total number of races to express the unknown quantities in terms of each other

step 2

Using the equation $24 = 4 + x + 2 + y$, solve for $x$ in terms of $y$: $x = 18 - y$

step 3

Combine the distances run in the last 12 races to express the unknown distances in terms of each other

step 4

Using the equation $4200 = 12/2 \times (4 \times 100 + x \times 200 + 2 \times 400 + y \times 800)$, solve for $x$ in terms of $y$: $x = -4y - 5/2$

step 5

Substitute $x = 18 - y$ from step 2 into the equation from step 4 to find the value of $y$

step 6

Solve the resulting equation to find the number of 800m races ($y$)

step 7

Substitute the value of $y$ back into the equation $x = 18 - y$ to find the number of 200m races ($x$)

[question 1] Answer

x = number of 200m races, y = number of 800m races

Key Concept

System of Equations

Explanation

To find the missing numbers of races, we set up a system of equations based on the total number of races and the total distance run in the last 12 races. We then solve this system for the unknowns.

Solution by Steps

step 1

Combine the known quantities of races and the total number of races to express the unknown quantities in terms of each other

step 2

Using the equation $24 = 4 + x + 2 + y$, solve for $x$ in terms of $y$: $x = 18 - y$

step 3

Combine the distances run in the last 12 races to express the unknown distances in terms of each other

step 4

Using the equation $4200 = 12/2 \times (4 \times 100 + x \times 200 + 2 \times 400 + y \times 800)$, solve for $x$ in terms of $y$: $x = -4y - 5/2$

step 5

Substitute $x = 18 - y$ from step 2 into the equation from step 4 to find the value of $y$

step 6

Solve the resulting equation to find the number of 800m races ($y$)

step 7

Substitute the value of $y$ back into the equation $x = 18 - y$ to find the number of 200m races ($x$)

[question 2] Answer

x = number of 200m races, y = number of 800m races

Key Concept

System of Equations

Explanation

To find the missing numbers of races, we set up a system of equations based on the total number of races and the total distance run in the last 12 races. We then solve this system for the unknowns.

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