Question

Three nos are in the ratio 3:4:5 . If the sum of the largest and the smallest equals the sum of the third and 52. Find the numbers ?

Answer 1

The numbers are 39, 52 and 65

Explanation:

The numbers are 3n,4n and 5n
We just need to find whether 3,4,5 or 6,8,10, or 9,12,15 etc
So 3n +5n= 4n+52
Simplify
8n=4n+52
Solve
4n=52
n=13
The 3 numbers are 39:52:65

Answer 2

39,52 and 65

Explanation:

There should be new triangle for propionate to 3:4:5
Let take x and multiple it to 3:4:5 to make new triangle
`3x+5x=4x+52`
`3x+5x-4x=52`
or
`4x=52`
or
`x=52/4`
or
`x=13`

Put the value of x =13 in `3x+5x=4x+52`
`3*13+5*13=4*13+52`
or
`39+65 = 52+52`

or
`104 = 104`

Hence the numbers are 39,52 and 65

Answer 3

39 : 52 : 65

`color(red)("There is ambiguity in this question.")`

Explanation:

Consider the ratios

We have 3 parts, 4 parts and finally 5 parts. This gives a total of 12 parts

Let the first number be `a`
Let the second number be `b`
Let the third number be `c`

Let the sum of all the numbers be `s`

So we have:

`a" : "b" : "c" " =" " 3" : "4" : "5`

3 parts < 4 parts < 5 parts so `" " a < b < c` and `a+b+c=s`
the first number is `a=3/12s`

the second number is `b=4/12s`

the third number is `c=5/12s`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Lets break down the wording of the question:

The sum of the largest and the smallest: `" "-> a+c`
equals:`" "->a+c=?`
the sum of:`" "->a+c=?+?`
the third:`" "->a+c=c+`
and 52: `" "->a+c=c+52`

`color(red)("This configuration points to "a=52)`

`color(green)("There is no point in continuing until this approach is confirmed as ok")`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~`color(magenta)("Possible error in the question")`~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(magenta)("The line:")`

`color(magenta)("the third: "->a+c=c+`

`color(green)("Should read:")`
`color(green)("the second: "->a+c=b+ )`

`color(green)("or")`
`color(green)("the middle: "->a+c=b+ )`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Solving for : "a+c=b+52)`

By substitution we have:

`3/12s+5/12s=4/12s+52`

`8/12s-4/12s=52`

`1/3s=52`

`=> s= 156`

`a=1/4xx156=39`
`b=1/3xx156=52`
`c=5/12xx156=65`