sincsenekdq
2022-09-13
Answered

Kathy stood on the middle rung of a ladder. She climbed up 5 rungs, moved down 2 rungs, and then climbed up 7 rungs. Then she climbed up the remaining 2 rungs to the top of the ladder. How many rungs are there in the whole ladder?

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Everett Melton

Answered 2022-09-14
Author has **12** answers

Number of rung of the ladder must be odd, because starts in the middle rung. The number of rungs on the ladder is some number, 2n+1 for some whole number 2.

The middle rung (t):

$$\frac{2n+1}{2}+\frac{1}{2}$$

The start $$t+5-2+7+2\phantom{\rule{0ex}{0ex}}=t+12\phantom{\rule{0ex}{0ex}}(\frac{2n+1}{2}+\frac{1}{2})+12=2n+1\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2n+1}{2}-2n=1-12-\frac{1}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2n+1-4n}{2}=\frac{-23}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow -2n+1=-23\phantom{\rule{0ex}{0ex}}\Rightarrow 2n-1=23\phantom{\rule{0ex}{0ex}}\Rightarrow 2n=24\Rightarrow n=12$$

Number of rungs $$(2n+1)=\phantom{\rule{0ex}{0ex}}2\times 12+1=25$$

There are 25 rungs on the ladder.

The middle rung (t):

$$\frac{2n+1}{2}+\frac{1}{2}$$

The start $$t+5-2+7+2\phantom{\rule{0ex}{0ex}}=t+12\phantom{\rule{0ex}{0ex}}(\frac{2n+1}{2}+\frac{1}{2})+12=2n+1\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2n+1}{2}-2n=1-12-\frac{1}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2n+1-4n}{2}=\frac{-23}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow -2n+1=-23\phantom{\rule{0ex}{0ex}}\Rightarrow 2n-1=23\phantom{\rule{0ex}{0ex}}\Rightarrow 2n=24\Rightarrow n=12$$

Number of rungs $$(2n+1)=\phantom{\rule{0ex}{0ex}}2\times 12+1=25$$

There are 25 rungs on the ladder.

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