mathematical problem?

Question

2 people are climbing a tower. 10mins pass and the first person is half way up the tower and the other a third. Between them they have climbed 160 steps. How many steps has the first person left to climb? (let the total number of steps in the tower be x)

This one has been bugging me for ages. Please give a very easibly understandable explaination and answer.

Answer 1

The first step is distinguish between useful and irrelevent information.

10 mins is the only time and there is no mention of speed, so we'll put that on one side for the moment.

The next two pieces of info relate two things (always a good starting point in maths problems)

It gets better. Half way up the tower and a third of the way = 160 steps.

You now have enough info to make an equation and they even give you a hint - to let total nuber of steps = x

1/2 times x + 1/3 times x = 160.

This is the critical step. Once you've got this you are home and dry.

Make denominators (underneath the line) the same.

Choose to express both 1/2 and 1/3 as 1/6s

(3/6 + 2/6)x = 160

5/6x = 160

multiply both sides by 6 and divide both sides by 5 to get x alone on one side of the equation

x = 160 times6/5

x = 192.......this is total number of steps in the tower.

so the man who's half way up has still to go half of 192 steps

=96 steps.

The one who is only 1/3 of the way up still has to climb 2/3 of 192 steps

=192 times 2/3 steps.....(note the relationship between already climbed 1/3 and needs to climb 2/3 - and 1/3 + 2/3 = 1)

= 64 times 2 steps

=128 steps to climb


The question didn't ask you the second part - I wanted you to have the slightly more difficult solution to use a a model for next time.

Answer 2

Well, if the total number of steps is x, and person A has climbed half of them (1/2 x) and person B has climbed a third of them (1/3 x) and between them they have climbed 160 steps, then:

1/2 x + 1/3 x = 160

rewriting so both fractions have a common denominator (the bottom number)

3/6 x + 2/6 x = 160
5/6 x = 160

dividing both sides by 5/6

x = 192

Therefore the total number of steps is 192. This means that person A has run up 96 steps and person B has run up 64 steps. We can check this easily

1/2(192) + 1/3(192) = 160
96 + 64 = 160

Answer 3

There are 192 steps in total therefore the first climber has 96 steps left to climb.

How to calculate:
climber a is 1/2 way up climber b is 1/3 - you must find a common denominator for the fractions therefore climber a has climbed 3/6 of the tower and climber b 2/6 so in total they have climbed 5/6 of the tower.

Divide 160 by 5 to find 1/6 = 32
Therefore 3/6 is equal to 96 and this is equivalent to half so he/she has another 96 to go.

Hope this makes sense

Answer 4

well, add the total distance climbed between the two : (1/2)x + (1/3)x = (3/6)x + (2/6)x = (5/6)x

this is the total distance climbed by the two, and we know this is equal to 160 steps:

(5/6)x=160

solving for x:

5x=960
x=192

so the total number of steps (x) is 192. since the first person has climbed halfway and must climb the other half of the tower, they must climb (192/2) steps, or 96 steps.

Answer 5

since the total number of steps is x, the 1st person has climbed x/2 steps and the second person has climbed x/3 steps. the key word in this problem is "between them". it says "Between them they have climbed 160 steps," which means that the sum x/2 +x/3 is equal to 160. it does NOT mean the two are 160 steps apart. you can now use the relationship to find x:
x/2 +x/3 =160
3x/6 +2x/6 =160
5x/6 =160
x= 6*160/5
=192
the tower has 192 steps, so the first person has 192/2 =96 steps left.

Answer 6

the 160 steps is 1/2 + 1/3 of the total tower.
this is 3/6+2/6.
so if you add the two climbers' distances together (so have 2 x the tower height...)
x=160*(6/3+6/2)
x=160*12/5
x=384. as the 2 climbers' heights have been added, the tower height is 384/2=192.
half of this is 96 - climber 1 still has this to go.
1/3 of it is 64, so climber 2 still has (64*2) 128 to go.

hope this helps.

Answer 7

x = total steps

1/2 x + 1/3 x = 160

3/6 x + 2/6 x = 160

5/6 x = 160

6/5 * 5/6 x = 6/5 * 160
x = 192

1/2 of x is 96, that many steps is how much the first climber still has to climb


<3 D

Answer 8

Let x = total number of steps.

1/2 x + 1/3 x = 160

3/6 x + 2/6 x = 160

5/6 x = 160

(6/5) 5/6 x = (6/5) 160
x = 192

1/2 of x is 96, which is how many the first climber still has left.

Answer 9

You need a fraction that represents both! SIXTHS WOULD WORK as 1/2 and 1/3 can be expressed as 3/6 and 2/6 resepctively! Then using 'sixths' then you have a common denominator as 3/6 = 50% and 2/6 = 33.3% so 3/6 + 2/6 = 160. 1/6 = 32. therefore .... 6/6 or the whole tower is 32*6 steps or 192! ..... nice one

Answer 10

Knowns/Unknowns
t = 10 min
Arate = ?
Brate = ?
x = ?

Equations
Arate(t) + Brate(t) = 160 steps
Arate(t) = x / 2
Brate(t) = x / 3

(x / 2) + (x / 3) = 160 steps

Multiply by (2)(3) = 6 to clear fractions

3x + 2x = 960 steps total

5x = 960 steps total

x = 192 steps total

Now take how far A went already (x/2) and subtract it from 192

192 - (192/2) = Steps Left
192 - 96 = Steps Left
96 = Steps Left

There you go, if you need explanation feel free to say so by adding a comment.