Reason with ratios and proportions

A ratio compares quantities by division. When a question says “the ratio of $A$ to $B$ is $a$ to $b$,” these three forms all say the same thing, in the same order:

The first number goes with $A$ and the second with $B$. Flip the fraction and you’ve flipped the comparison.

Say a club has $18$ seniors and $30$ juniors. Then

Juniors to seniors is

Both are true, but they answer different questions. The surest way to keep them apart is to put the words over the numbers: write “seniors” above the $3$ and “juniors” above the $5$ before you calculate anything.

Part-to-part and part-to-whole are different

The ratio $3:5$ compares seniors with juniors, one part of the club with another. That’s a part-to-part ratio. To get a fraction of the whole club, add the parts. The club has

equal parts in all, so seniors are $\frac38$ of the club and juniors are $\frac58$.

Now suppose a question says the ratio of seniors to all club members is $3:8$. This time the ratio already compares a part with the whole, so seniors are $\frac38$ of the club, as stated. Adding $3+8$ here would count the seniors twice and give the wrong fraction, $\frac3{11}$.

So before you add anything, check what the ratio compares:

• If it compares a part with the total, like seniors to all members, use it as stated.
• If its numbers count separate groups, like seniors and juniors, and the question wants a fraction of the combined whole or the total, add the parts.
• If the question compares one group with another, like juniors to seniors, keep the part-to-part ratio and add nothing.

Ratio, rate or unit conversion?

Ratio reasoning fits when the words give an ordered comparison, like “$a$ to $b$,” “for every,” “in the same ratio,” or one model-to-actual scale.

Two nearby kinds of question need different tools:

• When the comparison joins different units, like miles per hour or dollars per item, it’s a rate, which Use rates and unit rates covers.
• When a measurement has to pass through a chain of units, it’s a unit conversion, covered in Convert units with dimensional analysis.

Equivalent ratios, like $2:3$ and $4:6$, describe the same relationship at different sizes. If the ratio of $A$ to $B$ is $a:b$, you can write the actual amounts as

where $k$ is a positive scale factor, the number every ratio number gets multiplied by. Both amounts use the same $k$, because the relationship doesn’t change.

If $A:B=4:7$ and $A=28$, then the $4$ became $28$, so $k=7$. Multiply the second number by that same $7$:

When the ratio numbers count separate groups that make up a total $T$, add the parts:

So the first group is $\frac{a}{a+b}$ of the total and the second is $\frac{b}{a+b}$. That’s what you did with the conference passes: $11$ parts, $14$ passes in each.

A three-part ratio $a:b:c$ works the same way: