What is the ratio of certain inch measurements

Question

I'm responding to an answer I received from Arthur on 7-24-2014 (yes, I know was a long time ago.  I was unable to respond before now.  I apologize. 

 

In that answer. Arthur said his answer was based on the fact that he thought I meant area - when I had not said "area" at all.  I will repeat my question and try to make it very clear:

 

Line A is 5 inches long

Line B is 8 inches long

The ratio of Line A to Line B is 5:8

 

I have Line B and a new Line C

The ratio of Line B to Line C is 3:4

 

What is the ratio of Line A to Line C?

 

Answer 1

Hi Carol,

Here is another method for working your clarified problem.

Since each ratio is a simple fraction, just multiply the algebraic fractions to find A/C.

A/B = 5/8  and  B/C = 3/4,

so A/B * B/C = A/C or 5/8 *3/4 = 15/32

I am curious as to why you still wanted your question answered after so much elapsed time?  (It is not often that a student wants an answer long after any homework or test question is due!)

Answer 2

Defending Arthur, the problem originally said, "...WHAT IS THE RATIO OF TRIANGLE A TO C?"  That's not as clear as "What is the ratio of Line A to Line C?"  It might be still more clear to ask, "What is the ratio of the length of Line A to the length of Line C?"

 

Why so picky?  Well, lengths have a dimension:  inches, meters, miles, ...   Lines have equations that describe them; line segments have lengths, triangle sides are line segments.  So, Line Segment A is 5 inches long.  Yes, math is that precise.

 

I'll not reword your nice, clearer problem statement (because I think it is good enough to understand now even though I would not publish this wording in a math textbook).

 

A proportion, expressed as a:b, is a relationship of two values a and b such that there is a number x that makes ax+bx equal the total actual count of items (for example, total length in inches).

 

O.K., an example:  The ratio of tables to chairs is 1:4.  That means there is 1 table for each 4 chairs.  There might actually be 8 tables and 32 tables in the lunchroom.  This is because there is a value x=8 that makes the ratio 1:4 the same as 8:32.  We like to reduce ratios the same way we like to reduce fractions -- use the lowest possible pair of integers.

 

So, back to the clear statement of the problem.

  Ratio of lengths A:B is actually (because x=1) 5:8.

  Ratio of lengths B:C that is 3:4, but we are already know the actual length of B,

              which is why length of C=(4/3)*8.

    Well, that's not lowest whole integers, is it?  So, what value of x would help?

  The ratio of lengths B:C is better expressed as:

      B:C

      8: (4/3)8

      24:32                       (multiply both by 3, now divide both by 8)

      3:4             (that means that x=8/3 so that actual lengths of

                          8 and (4/3)*8 [that is, B:C] is 3:4.)

                [Note: start with 3:4; use x=8/3; get 8:(32/3) or (4/3)*8]

 

Finally, "What is the ratio of Line A to Line C?"  Well, the actual lengths are now known, so the ratio of the length of line segment A to the length of line segment C is:

     A:C

     5:(32/3)                 [oh, brother! those aren't the smallest integers, either!

     15:32                [but we can multiply by 3 because x=(1/3)]

         The ratio of A:C, or 15:32, has x=(1/3) such that the actual lengths are 5 and 32/3

              [you know how to change 32/3 to get length of C = 10 2/3 inches]

Answer 3

B : C :: 3 : 4                substitute

8 : C :: 3 : 4                means/extremes

32 = 3C                       divide both sides by 3

10 2/3 = C

 

A : C :: 5 : 10 2/3          change to improper fraction

A : C :: 5 : 32/3             divide

A : C :: 15 : 32