Please help with this worksheet

Note that the parts of one ratio always match the corresponding parts of the other ratio. That is, if one ratio is (first of first pair)/(first of second pair) then the parts of the other ratio will be (second of first pair)/(second of second pair).

If you keep the units with the measures, it is much easier to see when corresponding numbers are in corresponding places.

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1. You are given (small passenger count, small bus count) and (large passenger count, unknown bus count). The proportion can be written any of 4 ways, but the parts of each ratio need to correspond. The proportion of selection C is the only one that does. It has

(small passenger count)/(small bus count) = (large passenger count)/(unknown bus count)

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2. The pairs of interest are ...

(840 seats, 21 buses) and (unknown seats, 1 bus)

If we call these (x1, y1) and (x2, y2), then the proportion of selection A matches the form of the first proportion in the explanation above:

x2/y2 = x1/y1

(unknown seats) / (1 bus) = (840 seats) / (21 buses)

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3. The pairs of interest are ...

(36 salmon, 5 hours) and (unknown salmon, 12 hours)

I usually like to put the unknown in the numerator. I find it quicker to solve the proportion that way. Doing that, we can write ...

(36 salmon)/(5 hours) = (x salmon)/(12 hours)

If you multiply both sides of the equation by (hours)/(salmon), then the units cancel and you have just the numbers:

36/5 = x/12

(If you cannot cancel the units in that way, then there's something wrong with the proportion you have written.)

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4. Once you have written a proportion, you can rewrite it to any of the other three forms discussed in the explanation above. Here, the rewrite that answers the question is one that turns both ratios upside down. That is,

x/12 becomes 12/x
9/17 becomes 17/9

both ratios must be turned upside down in order for the solution to be the same. The result matches selection D. Effectively, you are multiplying the entire equation by (12·17)/(9x). (See comment below about inequalities.)

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5. You have the pairs ...

($25,200 dues, 3 months) and (unknown dues, 7 months)

Any of the ways of writing proportions discussed above will work. I find it convenient to put the unknown in the numerator on the left side of the equation. Doing that, we could have either of ...

x/7 = 25,200/3
x/25200 = 7/3

As stated above, there are several additional ways you can write the proportion. You can turn each of these upside down, and you can swap sides of the equal sign.

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Comment on inequalities

Sooner or later you may run across an inequality expressed using two ratios. If the variable you're solving for is in the numerator, then you can multiply by the inverse of its coefficient and you're done. If that happens to be negative (a very rare occurrence), then the direction of the inequality will have to be reversed according to the usual rule regarding negative multipliers.

It is important to pay attention to the requirement to do the same thing to both sides of the inequality. When multiplication or division is involved, the only rule you need to pay attention to is the one just mentioned regarding negative values.

2 > 1

-2 < -1 . . . . multiplication by a negative reverses the inequality direction

When you apply other functions, you need to be aware of what they do to ordering. The reciprocal function, for example, changes the ordering:

3 > 2

reciprocal(3) < reciprocal(2)

1/3 < 1/2 . . . . the reciprocal function reverses the order

You can avoid having to think about this by sticking to multiplication and division. Here, for example, if we divide the original inequality by 3·2 = 6 (a positive number), we get ...

3/6 > 2/6

1/2 > 1/3 . . . . the correct order

These ideas come into play when the variable of interest is in the denominator of a ratio involved in an inequality. Be aware. Be careful.