Easier way to divide a fraction by a fraction.

Say this is the problem:

$$\frac{3/8}{4/5}$$

As of right now, I would multiply both fractions by $40$ then simplify to get $\frac{15}{32}$

Or I would multiply $\frac{3}{8}$ and $\frac{5}{4}$ to get the same result.

Is there a better way to do this?

*In order for me to correctly ask this question, can someone leave a comment below on how to divide a fraction by a fraction using mathJax. Thanks

• Sorry just changed it. Commented Apr 8, 2016 at 18:41
• Both of your methods work, and are fairly simple. How much easier do you want it to be? Commented Apr 8, 2016 at 18:45
• No, it is fine. The better question is, how to simplify the fraction efficiently, see here. Commented Apr 8, 2016 at 18:46
• As for your typesetting question, you have a few options. \dfrac{(\frac{3}{8})}{(\frac{4}{5})} yields $\dfrac{(\frac{3}{8})}{(\frac{4}{5})}$ whereas \frac{3}{8}\div \frac{4}{5} yields $\frac{3}{8}\div \frac{4}{5}$. Alternatively still, one could use / to represent division as well, as Subhadeep Dey has done with his edit, or you could fiddle with spacing and not use parenthesis, e.g. $\dfrac{~~\frac{3}{8}~~}{~~\frac{4}{5}~~}$ Commented Apr 8, 2016 at 18:50
• Easier the multiplying by the reciprical? $\frac 3 8 * \frac 5 4= \frac{3*5}{8*4}$ isn't easy enough for you? Okay, you have to look out for common factors (which this one doesn't have) but sheesh how easy does this have to be? Commented Apr 8, 2016 at 19:00

I "look up". Quite literally.

I imagine you are one of the numbers, an then look up.

If the number sees one fraction line, or any odd number of them, then it belongs to the position where it will still see one fraction line above it (i.e., in the denominator)

If the number sees no fraction line, or two, or any even number of them, then it belongs to the position where it will still see no fraction line above it (i.e., in the numerator)

Note that wider lines hide the narrower lines above them

Example:

$\Large x=\frac{\frac{\frac{\frac{a}{b}}{c}}{d}}{\frac{\frac{\frac{e}{f}}{g}}{\frac{h}{i}}}$

Utterly confusing to simplify? Not really. It goes as:

• a sees 0 lines above it. Goes to the upper side of the final fraction.
• b sees 1 line above it. Goes to the down side of the final fraction.
• c sees 1 line above it (the line dividing a from b is narrower, so it can not be seen). Goes to the down side of the final fraction.
• d and e also sees 1 line. Down.
• f sees 2 lines. It goes to the upper side.
• g and h also both see 2 lines. Up.
• i sees 3 lines. That's the same as 1 line. Goes down.

So we get:

$x=\frac{a\,f\,g\,h}{b\,c\,d\,e\,i}$

We need to be careful when writing the fraction, so to have the correct line sizes - but we always do, it is not a special concern here.

The justification is simple. Each fraction line represents a "inverse of ", so every 2 of then cancel out.

• I like this. I'm not sure how much it would help a student who has yet to develop an intuitive understanding of division, but I still like it! Commented Apr 24, 2016 at 1:03
• Straight to the top answer. What we need is memory skills like this, not repeated derivation of the formula. Commented Apr 24, 2016 at 1:03
• How can I make the first fraction's font a little larger? (or someone do it, please) Commented Apr 24, 2016 at 1:03
• @zahbaz: I agree, one still needs to teach and practice and understand the why before, or along, with memorizing rules. Commented Apr 24, 2016 at 1:05
• Awesome answer. So in your example the length of each line confuses me a bit. Is the problem a over b divided by c over d? (The top part)? Commented Apr 24, 2016 at 1:57

A common way is to multiply the dividend by the reciprocal of the divisor:

$$\frac{3/8}{4/5} = \frac{3}{8} \div \frac{4}{5} = \frac{3}{8} \times \frac{5}{4} = \frac{3 \times 5}{8 \times 4} = \frac{15}{32}.$$

The other way you mention works as well:

$$\frac{3/8}{4/5} = \frac{\frac{3}{8} \times 40}{\frac{4}{5} \times 40} = \frac{\frac{120}{8}}{\frac{160}{5}} = \frac{15}{32}.$$

What may end up happening if you do it this way is that the fraction won't be in lowest terms yet and you'll need to factor the numerator and denominator to get the common factors out. (It is already in lowest terms in this case.) Doing it the first way gives you more of a chance to see common factors in the numerator and denominator that can be canceled since you've multiplied less together.

But they'll both get you there, and one isn't all that much harder than the other.

You remember the rule to multiply by the reciprical. You remember that the reciprical is just turning the dividing fraction "upside down". And you remember multiplying is multiplying the numerators together and the denominators together.

So

$$\frac{\frac 38}{\frac 45} = \frac 38 \cdot \frac 54 = \frac {3\cdot5}{8\cdot4}= \frac {15}{32}$$

That's pretty @@@@ing easy if you ask me. The only easier way I can think of is to ask someone else to do it for you. Which is probably actually significantly harder.

I don't know about easier but you could convert each fraction into an equivalent one such that both fractions have the same denominator. The denominators are the then irrelevant and the question is about counting.

To illustrate $$\frac {3}{8}=\frac {15}{40}\\ \frac{4}{5}=\frac {32}{40}$$

Now $$\frac {\frac {3}{8}}{\frac {4}{5}}=\frac {15}{40}÷\frac {32}{40}=15÷32=\frac {15}{32}$$

Notice the actual denominator isn't important if the fractions are the same type.

Not sure if this is easier or not but I thought it worth a mention.