Unit rates and Proportional Reasoning

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Scarlett

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This is a seventh grade math problem I am having trouble understanding and explaining. Could use some help. I know the answer is Alice and she wins by 1 yard, but not sure how to explain it.

Alice and Ann ran in a 100-yard dash. When Alice crossed the finish line, Ann was 10 yards behind her. The girls then repeated the race, with Alice starting 10 yards behind the starting line. If each girl ran at the same rate as before, who won the race? By how many yards?
 
Distance = Rate * Time

Alice #1 -- 100 yd = A1 * t1 <== We don't know the time or the rate.
Ann #1 -- 90 yd = n1 * t1 <== We don't know the rate, but the time is the same.

Solve each for t1

Alice #1 -- t1 = (100 yd)/A1
Ann #1 -- t1 = (90 yd)/n1

Substitution:

(100 yd)/A1 = (90 yd)/n1

Now we know how A1 and n1 are related.

You build the second race and see show us what you get.
 
Unit rates and proportional reasoning

Scarlett said:
This is a seventh grade math problem I am having trouble understanding and explaining. Could use some help. I know the answer is Alice and she wins by 1 yard, but not sure how to explain it.

Alice and Ann ran in a 100-yard dash. When Alice crossed the finish line, Ann was 10 yards behind her. The girls then repeated the race, with Alice starting 10 yards behind the starting line. If each girl ran at the same rate as before, who won the race? By how many yards?[

How did you arrive at the answer "1" ??? Please help, thank you/QUOTE]
Click to expand...
 
Hello, Scarlett!

This is a classic trick question.
The "obvious" answer is that they finish together, but that's wrong!

Alice and Ann ran in a 100-yard dash.
When Alice crossed the finish line, Ann was 10 yards behind her.
The girls then repeated the race, with Alice starting 10 yards behind the starting line.
If each girl ran at the same rate as before, who won the race and by how many yards?
Click to expand...

In the first race, Alice beat Ann by 10 yards.
When Alice ran 100 yards, Ann ran only 90 yards.
[That is, when Alice runs 10 yards, Ann runs only 9 yards.]


The second race is set up like this:

\(\displaystyle \begin{array}{ccccccc}\text{Alice} & * & --- & * & ---------- & * \\ && 10 && 100 \\ \\ &&\text{Ann} & * & ---------- & * \\ &&&& 100 \end{array}\)


Recall that: when Alice runs 100 yards, Ann runs only 90 yards.


Then their positions looks like this:

\(\displaystyle \begin{array}{cccccccc}\text{Alice} & * & --- & * & ------ & \bullet & --- & * \\ && 10 && 90 \\ \\ &&\text{Ann} & * & ------ & \bullet & --- & * \\ &&&& 90 \end{array}\)


Both of them have 10 yards to go.
But when Alice runs 10 yards, Ann runs only 9 yards.

Therefore, Alice also wins the second race ... by one yard.
 
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