Fraction problem

[Aion]

I was on a hiking trail, and after walking 7/12 of a mile, I was 5/9 of the way to the end. How long is the trail?

Point A: 5/9 of a certain length of x miles is 7/12 of a mile.

Point B: Exact determination of the value of x.

How do I go from point A to point B using only the facts known at the time of point A plus logical reasoning?

 

[MarkFL]

Your "Point A" translated into an equation becomes:

[MATH]\frac{5}{9}x=\frac{7}{12}[/MATH]
If we multiply both sides by 36, we get:

[MATH]20x=21[/MATH]
Can you finish?

 

[Aion]

Your "Point A" translated into an equation becomes:

[MATH]\frac{5}{9}x=\frac{7}{12}[/MATH]
If we multiply both sides by 36, we get:

[MATH]20x=21[/MATH]
Can you finish?

Yea this was exactly how I solved it. Thanks for the quick reply. My book doesnt show any solutions so I don't know if Im correct or not

 

[MarkFL]

Well, let's see if 7/12 is 5/9 of 21/20:

[MATH]\frac{7}{12}=\frac{5}{9}\cdot\frac{21}{20}=\frac{\cancel{5}}{3\cdot\cancel{3}}\cdot\frac{\cancel{3}\cdot7}{4\cdot\cancel{5}}=\frac{7}{12}\quad\checkmark[/MATH]
Looks like it checks out.

 

[Aion]

Well, let's see if 7/12 is 5/9 of 21/20:

[MATH]\frac{7}{12}=\frac{5}{9}\cdot\frac{21}{20}=\frac{\cancel{5}}{3\cdot\cancel{3}}\cdot\frac{\cancel{3}\cdot7}{4\cdot\cancel{5}}=\frac{7}{12}\quad\checkmark[/MATH]
Looks like it checks out.

Is it possible to solve this problem without multiplication?

 

[Aion]

Definition.
Let k/l be a nonzero fraction. Then for a number Q on the number line k/l of Q means the length of k concatenated parts when the segment [0,Q] is partitioned into l equal parts.

Theorem 1.4.

If k/l and m/n are fractions, then k/l of m/n = km/ln.

Proof. Since m/n = lm/ln, we see that [0,m/n] is lm copies of 1/ln. Therefore if we divide [0,m/n] into l equal parts, each part will be m copies of 1/ln, i.e., the length is km/ln. By the definition of k/l of m/n, we have proved theorem 1.4.

 

[Otis]

Is it possible to solve this problem without multiplication?

Why do you ask? Is that part of the exercise?

PS: When you have already solved an exercise and find yourself unable to check the answer, please say so up front. You ought to show your work when starting such threads, also, so that tutors can see what's going on. Please check out the forum's submission guidelines. Thanks!

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[Otis]

... k/l of m/n = km/(ln) ...

... lm/(ln) ... 1/(ln) ...

It doesn't seem like the definition and theorem in post #6 were written at the pre-algebra level. That notwithstanding, here's a note about typing algebraic ratios using a keyboard: Use grouping symbols around denominators containing more than one factor, as shown above in red, to eliminate ambiguity (eg: km/ln=km/l×n=kmn/l).

We use grouping symbols also with numerators/denominators containing more than one term. Cheers

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