Three numbers are in the ratio 2:3:4. The sum of their cubes is 0.334125. How do you find the numbers?

3 Answers
Jul 11, 2016

The 3 numbers are: 0.3, 0.45, 0.60.3,0.45,0.6

Explanation:

The question says there are three numbers but with a specific ratio. What that means is that once we pick one of the numbers, the other two are known to us through the ratios. We can therefore replace all 3 of the numbers with a single variable:

2:3:4 implies 2x:3x:4x2:3:42x:3x:4x

Now, no matter what we choose for xx we get the three numbers in the ratios specified. We are also told the sum of the cubes of these three numbers which we can write:

(2x)^3+(3x)^3+(4x)^3 = 0.334125(2x)3+(3x)3+(4x)3=0.334125

distributing the powers across the factors using (a*b)^c = a^c b^c(ab)c=acbc we get:

8x^3+27x^3+64x^3 = 99x^3 = 0.3341258x3+27x3+64x3=99x3=0.334125

x^3 = 0.334125/99 = 0.003375x3=0.33412599=0.003375

x = root(3)0.003375 = 0.15x=30.003375=0.15

So the 3 numbers are:

2*0.15, 3*0.15, 4*0.15 implies 0.3, 0.45, 0.620.15,30.15,40.150.3,0.45,0.6

Jul 11, 2016

The nos. are, 0.3, 0.45, and, 0.60.3,0.45,and,0.6.

Explanation:

Reqd. nos. maintain ratio 2:3:42:3:4. Therefore, let us take the reqd. nos. to be 2x, 3x, and, 4x.2x,3x,and,4x.

By what is given, (2x)^3+(3x)^3+(4x)^3=0.334125(2x)3+(3x)3+(4x)3=0.334125

rArr 8x^3+27x^3+64x^3=0.3341258x3+27x3+64x3=0.334125

rArr 99x^3=0.33412599x3=0.334125

rArr x^3=0.334125/99=0.003375=(0.15)^3...................(1)

rArr x=0.15

So, the nos. are, 2x=0.3, 3x=0.45, and, 4x=0.6.

This soln. is in RR, but, for that in CC, we can solve eqn.(1) as under :-

x^3-0.15^3=0 rArr (x-0.15)(x^2+0.15x+0.15^2)=0

rArr x=0.15, or, x={-0.15+-sqrt(0.15^2-4xx1xx0.15^2)}/2

rArr x=0.15, x={-0.15+-sqrt(0.15^2xx-3)}/2

rArr x=0.15, x=(-0.15+-0.15*sqrt3*i)/2

rArr x=0.15, x=(0.15){(-1+-sqrt3i)/2}

rArr x=0.15, x=0.15omega, x=0.15omega^2

I leave it to you to verify if complex roots satisfy the given cond. - hoping that you'll enjoy it!

Jul 11, 2016

Slightly different approach.

"First number: "2/9a->2/9xx27/20 = 3/10 -> 0.3

"Second number: "3/9a-> 3/9xx27/20=9/20->0.45

"Third number: "4/9a->4/9xx27/20=3/5->0.6

Explanation:

We have a ratio which is splitting the whole of something into proportions.

Total number of parts =2+3+4 = 9" parts"
Let the whole thing be a ( for all )

Then a=2/9a+3/9a+4/9a
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We are told that the sum of their cubes is 0.334125

Note that 0.334125 = 334125/1000000 -= 2673/8000

( aren't calculators are wonderful!)

So (2/9a)^3+(3/9a)^3+(4/9a)^3=2673/8000
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

8/729a^3+27/729a^3 +64/729a^3=2673/8000

Factor out the a^3

a^3(8/729+27/729 +64/729)=2673/8000

a^3=2673/8000xx729/99

a^3=19683/8000

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(brown)("Looking for cubed numbers")
Tony BTony B
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a^3 =(3^3xx3^3xx3^3 )/(10^3xx2^3)

Take the cube root of both sides

a=(3xx3xx3)/(10xx2) = 27/20

color(white)(2/2)

color(brown)("So the numbers are:")

"First number: "2/9a->2/9xx27/20 = 3/10 -> 0.3

"Second number: "3/9a-> 3/9xx27/20=9/20->0.45

"Third number: "4/9a->4/9xx27/20=3/5->0.6