Question

Kathy stood on the middle rung of a ladder. She climbed up 5 rungs, moved down 3 rungs, and then climbed up 6 rungs. Then she climbed up remaining 4 rungs to the top of the ladder. How many rungs are there in the whole ladder?

There are——— rungs on the ladder.

Answer 1

Explanation:

middle rung of a ladder. that means the ladder has an odd number of rungs.

let's say there are 7 rungs. the middle rung is the 4th rung (3 down and 3 up from there). this is not possible for an even number of rungs. e.g for 6 rungs, if we divide by 2 (= 3), when we stand on the third rung, there are 3 up but only 2 down.

so, the ladder has x rungs. x is an odd number.

therefore, the middle rung is (x + 1)/2.

now, based on the rung number (x + 1)/2 she added 5, subtracted from that 3, and then added 6 to all that. and then she added 4 to reach the top. that means that equals x.

that expression is

(x + 1)/2 + 5 - 3 + 6 + 4 = (x + 1)/2 + 12

and that is equal to x itself :

(x + 1)/2 + 12 = x

this is now an equation for x we can solve. we start by removing the fraction by multiplying everything by 2 :

(x + 1) + 24 = 2x

x + 1 + 24 = 2x

x + 25 = 2x

x = 25

there are 25 rungs on the ladder.