Ratios, Proportions, and Conversions

ratio problem will involve some form of scaling (usually in size or amount), or the rate at which something is done.


If Johnny can bike five blocks in twelve minutes, how many blocks can he bike in sixty minutes?

Strategy for ratios:

  1. Convert all corresponding values to the same units.
  2. Set up the one ratio that the problem provides. Be sure to determine which category is going in the numerator and which category is going in the denominator. Either way will work, but you have to be consistent when you make your second ratio.
  3. Set up your second ratio with what you are solving for represented by the variable x.
  4. Set the ratios equal to each other and solve.

Ratio: A relationship that compares the relative size of two amounts.  Ratios can be written with colons (\(4:7\)), as a fraction (\(\frac47\)), or using words (4 to 7).

Ratios can be used to compare parts to wholes, or parts to other parts.  For example, let’s say you make a salad using 1 cucumber, 3 carrots, and 2 heads of lettuce.  A part to whole ratio could be the ratio of cucumbers to all the vegetables, which would be 1 : 6.  A part to part ratio could be the ratio of carrots to heads of lettuce, which would be 3 : 2.

Proportion:  Two ratios set equal to each other.  Proportions are useful in certain solutions, but be sure to watch your units!  Write your proportion in words first, to safeguard against any potential unit errors.  To solve a proportion, cross multiply and then use standard operations.

$$\frac pq=\frac rs,$$ therefore, $$p \;\times\;s\;=\;q\;\times\;r$$

Converting units is an important skill to know for the SATs.  View this skill as building a conversion ladder, with the goal of piecing together a pathway to the proper unit.  In the example below, we will piece together a path that converts 2 gallons, into cups.


There are four quarts in a gallon, 2 pints in a quart, and 2 cups in a pint.  All we had to do was multiply the fractions across.  

Note: Don’t be nervous if you don’t know the relationships between these units (i.e. the number of pints in a quart); the SAT will provide this information when it’s needed.