Answer:
Step-by-step explanation:
The topic of proportions has some specific terminology that you may need to know. For instance, given the following proportion equation:
\small{ \dfrac{a}{b} = \dfrac{c}{d} }
b
a
=
d
c
...the values in the "b" and "c" positions are called the "means" of the proportion, while the values in the "a" and "d" positions are called the "extremes" of the proportion.
A basic defining property of any proportion is that the product of the means is equal to the product of the extremes. In other words, given the proportional statement:
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Proportions
\small{ \dfrac{a}{b} = \dfrac{c}{d} }
b
a
=
d
c
...we know that it must be true that ad equals bc. This fact about proportions is, in effect, the cross-multiplication demonstrated on the previous page. And this cross-multiplication fact about the products of the means and extremes is occasionally turned into a homework problem, such as:
Is \small{ \bm{\color{green}{ \dfrac{24}{140} }}}
140
24
proportional to \small{ \bm{\color{green}{ \dfrac{30}{176} }}}
176
30
? Explain why (or why not) without simplifying the fractions.
For these fractions (that is, these ratios) to be proportional (that is, for them to create a true proportional equation when they are set equal to each other), it has to be true that the product of the means of that equation is equal to the product of the extremes. So I can figure out if the two fractions are indeed proportional to each other (without simplifying them) by finding these two products.
In other words, by specifying that I'm supposed to not simplify the fractions, they are hinting that they are wanting me to find the product of 140 and 30 (being the means, if I keep the fractions in the same order as they've given them to me) and the product of 24 and 176 (being the extremes), and then see if these products are equal. So I'll check:
140 × 30 = 4200
24 × 176 = 4224
While these values are close, they are not equal, so I know the original fractions cannot be proportional to each other. So my answer is:
The fractions are not proportional because the product of their means does not equal the product of their extremes.
If I'd reversed the fractions, and used 176 and 24 as my means and 30 and 140 as my extremes, I would have gotten the same products (just in reverse order), and thus the same answer (namely, that the fractions are not proportional). So don't worry about which fraction is "first" or "second"; either way will work.
Is \small{ \bm{\color{green}{ \dfrac{42}{55} }}}
55
42
proportional to \small{ \bm{\color{green}{ \dfrac{50}{65} }}}
65
50
? Justify your answer without reducing the fractions.
To confirm proportionality (or to disprove it), I'll need to set up the proportion, multiply the means, multiply the extremes, and compare the results. Or, which is the same thing (but without doing an equation that might not actually be true), I'll multiply one fraction's denominator by the other's numerator, and vice-versa:
(42)(65) = 2,730
(55)(50) = 2,750
Once again, they're close, but they're not equal. So:
The fractions are not proportional because the product of their means does not equal the product of their extremes.
Are \small{ \bm{\color{green}{ \dfrac{42}{273} }}}
273
42
and \small{ \bm{\color{green}{ \dfrac{170}{1105} }}}
1105
170
proportional? Explain your answer without reducing the fractions or converting to a common denominator.
I "cross-multiply" (meaning, in this context, multiplying one fraction's numerator by the other's denominator, and vice versa):
(42)(1,105) = 46,410
(273)(170) = 46,410
Finally, a pair that is proportional!
The fractions are proportional because, when set up as a proportion, the product of the means is equal to the product of the extremes.
