Question 1206954
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Another tutor had answered this previously
<a href="https://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-and-ratio-word-problems.faq.question.37859.html">https://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-and-ratio-word-problems.faq.question.37859.html</a>


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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<ul><li>1st height = 48y = 48*(1/12) = 4 ft</li><li>2nd height = 10y = 10*(1/12) = 5/6 of a ft = 10 inches</li><li>3rd height = 120y = 120*(1/12) = 10 ft</li><li>4th height = 3y = 3*(1/12) = 1/4 of a ft = <font color=red size=4>3 inches</font> is her shortest height.</li></ul>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:<font color=red>3</font> makes it immediately clear that <font color=red>3 inches</font> 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: <font color=red size=4>3 inches</font> (aka 1/4 of a foot)
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