Ratio Problem from PowerPrep

[giddy]

The jewels in a crown consist of diamonds, rubies, and emeralds. If the ratio from diamonds to rubies is 5/6 and the ratio from rubies to emeralds is 8/3, what is the least number of jewels that can be in a tiara?

16
22
40

SPOILER: 53

67


I got the answer correctly but I guessed the answer. I tried to solve it but I didn't come up with the understandable solution. Your help is appreciated. Thank you.

[Oldman]

hmmm...seems like you need 3 equations...but only enough info is given for two equations? Was this in powerprep? I don't remember this one.

[Goldust]

Since the number of rubies is common to both ratios, try to find the LCM of the rubies factor is each ratio i.e. LCM of 6 and 8. This is 24. Hence, there must be 24 rubies in the least number of jewels. Solve the ratios to get 20 diamonds and 9 emeralds. Adding up all the jewels, you get 53.

[giddy]

Originally Posted by Oldman

hmmm...seems like you need 3 equations...but only enough info is given for two equations? Was this in powerprep? I don't remember this one.

I just reviewed the PP test I took 2 days ago. The question is correct.

[giddy]

Originally Posted by Goldust

Since the number of rubies is common to both ratios, try to find the LCM of the rubies factor is each ratio i.e. LCM of 6 and 8. This is 24. Hence, there must be 24 rubies in the least number of jewels. Solve the ratios to get 20 diamonds and 9 emeralds. Adding up all the jewels, you get 53.

Thanks, Goldust. The only way I was solving it that was making sense to me was that I multiplied 6 and 8 and added 5. But I left out 3 so that won't work. Your explanation was greatly appreciated. Thank you.

[Oldman]

Makes sense now, thanks Goldust! Thanks for posting giddy, I had never seen that problem.

[mystic87]

easiest way to find the LCM is to find the prime factors of each of the numbers and multiply what they don't have. So 6=2X3 and 8 = 2x2x2. So you need to multiply 2 to 6 and 3 to 8 to get their LCM 24.

Since its 5 Diamonds:6 Rubies, you have to multiply both by 4
and multiply 3 to 8 Rubies: 3 Emeralds.

Add them all together, you'll get 53

[sandipank]

Originally Posted by giddy

Thanks, Goldust. The only way I was solving it that was making sense to me was that I multiplied 6 and 8 and added 5. But I left out 3 so that won't work. Your explanation was greatly appreciated. Thank you.

You can solve this problem in this way also...
D:R=5:6 and R:E=8:3
now when taking composite ratio(I forgot the exact term) for D:R:E=5*8:6*8:3*6=40:48:18=20:24:9(Make the common term i.e. Ruby as 48 by taking LCM). so min no. of jewels =53