Ratios and Rates

Ratios and Rates

A ratio compares two quantities of the same kind — for example, the ratio of boys to girls in a class. A rate compares two quantities of different kinds — for example, miles per hour compares distance to time. Both show up constantly in competition problems.

Ratio Notation

The ratio of a to b can be written as a:b, as \frac{a}{b}, or in words as "a to b." All three mean the same thing. A ratio like 3:5 doesn't tell you the exact amounts — only that for every 3 units of the first quantity, there are 5 units of the second. The actual amounts could be 3 and 5, or 30 and 50, or any other pair in the same proportion.

Simplifying Ratios

Just like fractions, ratios should be simplified by dividing both parts by their greatest common factor. The ratio 18:24 simplifies to 3:4 after dividing both by 6. Ratios with three or more parts simplify the same way: 12:18:30 simplifies to 2:3:5 after dividing everything by 6.

Working With "Parts"

A ratio splits a total into equal-sized parts. If a ratio is 3:5, think of the whole as 3+5=8 parts. If a total of 40 is split in the ratio 3:5, each "part" is worth 40 \div 8 = 5, so the two pieces are 3 \times 5 = 15 and 5 \times 5 = 25. This "total parts" trick is one of the fastest ways to handle ratio problems on contests.

Unit Rates

A unit rate expresses a rate with a denominator of 1, such as "miles per hour" or "dollars per item." To find a unit rate, divide the first quantity by the second. If a car travels 180 miles in 3 hours, its unit rate is \frac{180}{3} = 60 miles per hour. Unit rates make it easy to compare deals: a \$12 pack of 8 pens costs \$1.50 per pen, while a \$20 pack of 16 pens costs \$1.25 per pen — so the second pack is the better deal per pen.

Example

A recipe calls for flour and sugar in the ratio 5:2. If a baker uses 35 cups of flour, how much sugar is needed?

Step 1: Set up the ratio as a proportion, keeping flour and sugar in the same order on each side: \frac{5}{2} = \frac{35}{s}

Step 2: Notice 35 = 5 \times 7, so the scale factor from the flour side of the ratio to the actual amount is 7.

Step 3: Apply the same scale factor to sugar: s = 2 \times 7 = 14.

So the baker needs 14 cups of sugar.

Key Takeaways

• A ratio compares quantities of the same kind; a rate compares different kinds of quantities.
• Simplify ratios by dividing all parts by their greatest common factor.
• Think of a ratio as splitting a total into equal "parts" — divide the total by the sum of the ratio parts to find the value of one part.
• A unit rate has a denominator of 1 and is great for comparing deals or speeds.