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Question 1206954: Alice changes size several times. The ratio of her original height to her second height is 24 to 5.
The ratio of her second height to her third height is 1 to 12. The ratio of her original height to her fourth height is 16 to 1.
The tallest of these four heights is 10 feet. What is her shortest height? (Source: MATHCOUNTS)
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Another tutor had answered this previously
https://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-and-ratio-word-problems.faq.question.37859.html
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Here's another approach.
1st:2nd = 24:5
2nd:3rd = 1:12 which is equivalent to 5:60 after multiplying both parts by 5
24:5 and 5:60 combine to 24:5:60
The ratios of the 1st four heights are 24:5:60:x
1st:4th = 16:1 = 24:x
16:1 = 24:x
16/1 = 24/x
16x = 1*24
x = 24/16
x = 3/2
24:5:60:x updates to 24:5:60:3/2
Double each part of that ratio to get 48:10:120:3
IF Alice had a tallest height of 120 ft, then the ratio 48:10:120:3 would represent the first four heights.
However, her tallest height is 10 ft instead. So we need to scale this ratio to get 48y:10y:120y:3y where y is some positive real number.
The tallest height is 120y, which is 10 ft, so,
120y = 10
y = 10/120
y = 1/12
Then we have these heights- 1st height = 48y = 48*(1/12) = 4 ft
- 2nd height = 10y = 10*(1/12) = 5/6 of a ft = 10 inches
- 3rd height = 120y = 120*(1/12) = 10 ft
- 4th height = 3y = 3*(1/12) = 1/4 of a ft = 3 inches is her shortest height.
Side note: multiply by 12 to convert from feet to inches.
A shortcut is to realize that 120 inches = 10 ft, then the ratio 48:10:120:3 makes it immediately clear that 3 inches is her shortest height.
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Check:
1st:2nd = 24:5
4:5/6 = 24:5
4/( 5/6 ) = 24/5
4*(6/5) = 24/5
24/5 = 24/5
This confirms that the first two heights (4 ft and 5/6 ft) are in ratio 24:5
2nd:3rd = 1:12
5/6:10 = 1:12
(5/6)/10 = 1/12
5/60 = 1/12
1/12 = 1/12
This confirms that the 2nd and 3rd heights (5/6 ft and 10 ft) are in ratio 1:12
One last thing to check.
1st:4th = 16:1
4:1/4 = 16:1
4/(1/4) = 16/1
4*4 = 16/1
16/1 = 16/1
This confirms that the 1st and 4th heights (4 ft and 1/4 ft) are in ratio 16:1
All three ratio requirements have been met. The answer is fully confirmed.
If the fractions seem a bit messy, then converting each measurement in feet to inches would help avoid some fractions to make things look cleaner.
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Answer: 3 inches (aka 1/4 of a foot)
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