My daughter is stuck on this question and we're not sure what the answer is. I'm rusty with math skills, so I don't understand how to help her, or what approaches she's tried. Please help us! 10.5/12 and 2/5
13/7 and 7/13
9/30 and 1.5/5
7/5 and 10/8
$\endgroup$ 35 Answers
$\begingroup$You're really asking which pairs of ratios are equal. In general $a/b=c/d$ when $ad=bc$. In this case, the only ratio that satisfies this is $9/30=1.5/5$, since $9\times5=45=1.5\times30$.
$\endgroup$ $\begingroup$There are diverse ways to answer this question. First, let me recall $\;\dfrac ab=\dfrac cd\;$ means, loosely speaking, that ‘$a$ is to $b$ as $c$ is to $d$’. A more rigourous phrasing is that $\;\dfrac ab=\dfrac cd\;$ means that the products $a\times d=b\times c$.
The first phrasing lets you see at once the first pair of ratios is not a proportion, since $10.5/12$ is close to $1$, while $2/5<1/2$.
The second pair is not, too: actually $13/7$ and $7/13$ are inverses of each other.
The third ratios do form a proportion: indeed $\;30=\color{red}{6}\times 5$ and $\;\color{red}6\times 1.5=9$.
The fourth ratios do not: a quick way to see this is to note $\;\dfrac75=1+\dfrac25$, while $\;\dfrac{10}8=1+\dfrac18$.
$\endgroup$ $\begingroup$Two numbers $a,b$ are proportional when there is an integer $A \neq 0$ and such that $a\cdot A = b$ or $ a = A \cdot b$.
You can check that the only pair of numbers that you have listed satysfying the above requirement are $\frac{9}{30}$ and $\frac{1.5}{5}$ because $\frac{9}{30} = 1 \cdot \frac{1.5}{5}$.
$\endgroup$ 1 $\begingroup$They are asking which pair of fractions are equal. To check this, you will need to convert the fractions so that they have a common denominator.
Here is an example of getting a common denominator for the first pair of fractions (recall that multiplying by fractions like $\frac{7}{7}$ or $\frac{12}{12}$ does not change a number's value since these are equal to $1$):
$$\frac{10.5}{12} \stackrel{?}{=} \frac{2}{5}$$
$$\frac{10.5}{12} \times \frac{5}{5} \stackrel{?}{=} \frac{2}{5} \times \frac{12}{12}$$
$$\frac{52.5}{60} \neq \frac{24}{60}$$
Now it is obvious that these two fractions are not the same.
Another example:
$$\frac{9}{30} \stackrel{?}{=} \frac{1.5}{6}$$
$$\frac{9}{30} \times \frac{6}{6} \stackrel{?}{=} \frac{1.5}{6} \times \frac{30}{30}$$
$$\frac{45}{180} = \frac{45}{180}$$
Here the fractions are indeed equal.
As Barry mentioned, you can use the shortcut to check if $ad \stackrel{?}{=} bc$ since this amounts to checking if the numerators are the same after the denominators have been made equal.
$\endgroup$ $\begingroup$A proportion is just a statement that two ratios (i.e. fractions) are equal. So, for example, $\frac{1}{2}$ and $\frac{2}{4}$ form a proportion since $\frac{2}{4} = \frac{1}{2}$ (you can see this by reducing $\frac{2}{4}$).
One way to check to see if two fractions form a proportion is to "cross-multiply" them. In the example I just gave, we multiply the numerator of the first fraction by 4, and the numerator of the second fraction by 2, and compare the two sides: \begin{gather*} \frac{1}{2} \stackrel{?}{=} \frac{2}{4} \\ 4 \cdot 1 \stackrel{?}{=} 2 \cdot 2\\ 4 \stackrel{?}{=} 4 \end{gather*} Since the equality we get at the end is true, the fractions do, indeed, form a proportion.
Contrast that with the fractions $\frac{1}{3}$ and $\frac{3}{4}$. Clearly these are not equal, so they don't form a proportion, but let's check using the method we just learned: \begin{gather*} \frac{1}{3} \stackrel{?}{=} \frac{3}{4} \\ 4 \cdot 1 \stackrel{?}{=} 3 \cdot 3 \\ 4 \stackrel{?}{=} 9 \end{gather*} Four is certainly not equal to nine, so the fractions are not in proportion.
I'd be happy to talk a little more about why this procedure works, if you would like, but I hope that this is at least enough to get you started. Good luck, and have fun!
$\endgroup$
