3.2 Ratios, Proportions, and Unit Rates

Understanding Ratios and Proportions

Ratios and proportions are among the most frequently tested concepts on the SHSAT. They form the basis of word problems involving mixtures, scaling, speeds, rates, and unit pricing. A clear conceptual grasp of these topics allows students to solve multi-step problems efficiently without a calculator.

A ratio is a comparison of two or more quantities by division. Ratios can be expressed in three ways:

1. Using the word "to": 3 to 4
2. Using a colon: 3 : 4
3. As a fraction: 3/4

Ratios can describe part-to-part or part-to-whole relationships. For example, if a bag contains 12 blue marbles and 8 red marbles, the part-to-part ratio of blue to red marbles is 12 : 8, which simplifies to 3 : 2. The part-to-whole ratio of blue marbles to total marbles is 12 : (12+8) = 12 : 20, which simplifies to 3 : 5. Distinguishing between these two types of ratios is critical on the SHSAT, as questions often give a part-to-part ratio but ask for a part-to-whole calculation.

Three-Part Ratios

The SHSAT occasionally features three-part ratios, such as x : y : z = 2 : 3 : 5. In these problems, you can define the actual quantities in terms of a common multiplier, k. Let x = 2k, y = 3k, and z = 5k. If the sum of these quantities is 120, we can set up the equation: 2k + 3k + 5k = 120, which simplifies to 10k = 120, leading to k = 12. Thus, x = 2(12) = 24, y = 3(12) = 36, and z = 5(12) = 60.

Proportions and Cross-Multiplication

A proportion is an equation stating that two ratios are equal: a/b = c/d To solve for an unknown variable in a proportion, use cross-multiplication: a * d = b * c For example, if x/15 = 4/6, cross-multiplying yields: 6x = 60, which simplifies to x = 10. Always simplify ratios before cross-multiplying to make the calculations easier to manage without a calculator. For instance, simplify 4/6 to 2/3 first: x/15 = 2/3, which cross-multiplies to 3x = 30, leading to x = 10.

Direct vs. Inverse Proportionality

Understanding the relationship between two variables is essential for setting up the correct equation.

Direct Proportion

In a direct proportion, as one quantity increases, the other increases at a constant rate. The ratio between the two variables remains constant: y/x = k, which can be written as y = kx, where k is the constant of proportionality. Example: If 5 notebooks cost $12.50, the cost is directly proportional to the number of notebooks. 12.50 / 5 = 2.50, meaning each notebook costs $2.50. If you buy 8 notebooks, the cost is 8 * 2.50 = $20.00.

Inverse Proportion

In an inverse proportion, as one quantity increases, the other decreases proportionally. The product of the two variables remains constant: x * y = k, which can be written as y = k/x, where k is the constant. Example: The time it takes to paint a fence is inversely proportional to the number of workers. If 3 workers take 6 hours to paint a fence, the product is constant: 3 workers * 6 hours = 18 worker-hours (k = 18). If 2 workers paint the same fence, the time required is: 2 workers * t = 18, leading to t = 9 hours.

Direct vs. Inverse Comparison Table

Unit Rates and Unit Pricing

A rate is a ratio comparing two quantities with different units, such as miles per hour or price per ounce. A unit rate is a rate with a denominator of 1. To find a unit rate, divide the first quantity by the second quantity. For example, if a car travels 180 miles in 3 hours, the unit rate is: 180 miles / 3 hours = 60 miles per hour (mph).

Unit Price Comparisons

Unit pricing problems ask you to find which item is the "best buy." Suppose Brand A sells a 12-ounce box of cereal for $3.60, and Brand B sells an 18-ounce box for $5.04.

• Brand A unit price: 3.60 / 12 = 0.30 dollars per ounce.
• Brand B unit price: 5.04 / 18 = 0.28 dollars per ounce. Brand B has a lower unit price and is the better value.

Scale Drawings and Scale Factors

A scale drawing represents a real-world object with dimensions scaled down or up by a constant ratio, known as the scale factor. If a map scale is 1 inch = 15 miles, then a distance of 4.5 inches on the map represents: 4.5 inches * (15 miles / 1 inch) = 67.5 miles.

Area Scaling

A major trap on the SHSAT involves area and volume changes in scale drawings. If the linear dimensions of a drawing are scaled by a factor of k, the area is scaled by a factor of k^2. Example: A rectangular garden has actual dimensions of 20 feet by 30 feet, for an area of 600 square feet. On a scale drawing with a scale of 1 inch = 10 feet, the scale factor is k = 1/10. The dimensions in the drawing are 2 inches by 3 inches, giving an area of 6 square inches. Notice that the ratio of drawing area to actual area is: 6 / 600 = 1/100 = k^2. Do not use the linear scale factor when calculating scaled areas or volumes; always square the scale factor for area and cube it for volume.