Ratios, rates, proportions and units on the SAT

Example 1: a proportion

Problem A recipe uses 3 cups of flour for 24 cookies. How many cups of flour are needed for 60 cookies?

1. Set flour over cookies equal in both cases. Call the unknown amount of flour f.
   $$\frac{3}{24} = \frac{f}{60}$$
2. Cross-multiply.
   $$24f = 180$$
3. Divide by 24.
   $$f = 7.5$$

Answer 7.5 cups

Example 2: a chain of unit conversions

Problem A car travels 66 feet per second. What is its speed in miles per hour? (1 mile = 5,280 feet)

1. Seconds to hours: there are 3,600 seconds in an hour, so multiply by $\frac{3600 \text{ s}}{1 \text{ h}}$.
   $$66 \times 3600 = 237600$$
2. That is 237,600 feet per hour. Feet to miles: divide by 5,280.
   $$\frac{237600}{5280} = 45$$

Answer 45 miles per hour

Example 3: ratio parts

Problem A jar holds red and blue marbles in the ratio $4 : 7$. There are 132 marbles. How many are blue?

1. Count the parts.
   $$4 + 7 = 11$$
2. Find one part.
   $$\frac{132}{11} = 12$$
3. Blue is 7 parts.
   $$7 \times 12 = 84$$
4. Check: red is $4 \times 12 = 48$, and $48 + 84 = 132$.

Answer 84 blue marbles

Example 4 (SAT-hard): area units

Problem A rectangular room is 4 yards by 5 yards. Carpet costs $3.50 per square foot. How much does it cost to carpet the room?

1. Area in square yards.
   $$4 \times 5 = 20$$
2. Convert to square feet. One square yard is $3 \times 3 = 9$ square feet.
   $$20 \times 9 = 180$$
3. Multiply by the price per square foot.
   $$180 \times 3.5 = 630$$
4. The trap answer multiplies by 3 instead of 9: $60 \times 3.50 = 210$.

Answer $630