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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

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  • $\begingroup$ Would any of you happen also to know what it is for these numbers? $\endgroup$ – awesomesauce Aug 11 '15 at 14:41
  • $\begingroup$ 2.5/9 and 5/18 1/3 and 7/9 3/16 and 2/11 6.3/7 and 7/6.3 $\endgroup$ – awesomesauce Aug 11 '15 at 14:42
  • $\begingroup$ Oh, and thank you guys! $\endgroup$ – awesomesauce Aug 11 '15 at 14:46
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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 $9\times5=45=1.5\times30$.

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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 $\;\dfrac{10}8=1+\dfrac18$.

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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 $\frac{9}{30} = 1 \cdot \frac{1.5}{5}$.

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    $\begingroup$ Higher Algebra by Hall and Knight defines it thus: "When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if a/b = c/d, then a,b,c,d are proportionals. [ Bold facing mine ] $\endgroup$ – true blue anil Aug 11 '15 at 14:47
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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.

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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!

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