Which value should be in the numerator? How to think about fractions? **Updated**

Question

If I have a word problem with a fraction, how should I know which value should be in the numerator? Say I have the following problem:

"You stand 180 meters away from a radio tower. One wavelength is 260 meters. How many wavelengths away from the radio tower are you?"

My picture:

I thought of this as an equation which I have to "balance", i.e. \begin{align} 180x &= 260 \\ x&= \frac{260}{180}= 1.45 \tag 1 \end{align} ... but this was wrong. In the correct answer, the numerator and denominator are flipped, i.e. \begin{align} x&= \frac{180}{260}= 0.69 \tag 2 \end{align}

How should I know which value should be in numerator? How should I think?

Update:

Great answers!

My confusion arised beacuse $\frac{180}{260}$ was correct instead of $\frac{260}{180}$. So suppose the question is instead formulated as:

"You stand 260 meters away from a radio tower. One wavelength is 180 meters. How many wavelengths away from the radio tower are you?"

But in this case I assume $x=\frac{180}{260}$ is wrong, and the correct answer is instead: \begin{align} 180x &= 260 \\ x&= \frac{260}{180}= 1.45 \tag 3 \end{align}

I.e. here $\frac{260}{180}$ is correct but in the first formulation $\frac{180}{260}$ was correct. How should I think so I not confuse the numerator/denominator in different cases?

Answer 1

Note the question carefully:

"How many wavelengths away from the radio tower are you?"

Clearly, it wants you to find out the number of wavelengths. Now you may proceed as:

Suppose the number of wavelengths is $x$.

Can you take it up from here?

Then the length of $x$ wavelengths in all is $260x$. Thus, to cover $180$m, we should have $$260x=180\implies x=\frac{180}{260}=\frac{9}{13}\approx0.69$$

Update: Now for the question which you added later...

Similar logic here also. The length of a wavelength is $180$m. Say the number of wavelengths required to cover $260$m is $y$.

Then think what the total length of all the $y$ wavelengths taken together will be and equate it with your distance from the tower.

The total length of all the $y$ wavelengths taken together is $180y$. Thus, to cover $260$m, the number of wavelengths required will be $$180y=260\implies y=\frac{260}{180}=\frac{13}{9}\approx1.44$$

Answer 2

This is known as a problem in dimensional analysis.

$$180 ~\text{meters} \times \frac{1 ~\text{wavelength}}{\text{260 meters}} ~=~ \frac{180}{260} ~\text{wavelengths}.$$

To illustrate with a more familiar example, suppose that you drive for $(2)$ hours, at a speed of $(60)$ miles per hour. How far have you driven?

$$2 ~\text{hours} \times \frac{60 ~\text{miles}}{\text{1 hour}} ~=~ (2 \times 60) ~\text{miles}.$$

Answer 3

It's possible that a certain amount of confusion with fractions arises because the terms "numerator" and "denominator" aren't commonly used in other contexts. In a fraction $\frac{p}{q}$:

• The denominator $q$ specifies "how many things" make one unit, i.e., it denominates the value of $\frac{1}{q}$.

• The numerator $p$ specifies "how many units" $\frac{1}{q}$ there are in the fraction, i.e., it counts or enumerates portions of size $\frac{1}{q}$.

Here, the "unit of stuff" is one wavelength. There are $q = 260$ meters in one wavelength, so one meter is $\frac{1}{260}$ wavelengths. If we want to know what fraction of a wavelength is $180$ meters, we take $180$ one-meter units, so $p = 180$, hence $\frac{p}{q} = \frac{180}{260}$ wavelengths.