Section 2.6: Ratios and Proportional Relationships

Ratios, Rates, and Unit Rates

Competency 0002 (Ratios and Proportional Relationships and Number Systems) is 30% of Part Two, the single largest math strand on Field 222, so proportional reasoning deserves dedicated study.

A ratio compares two quantities of the same or different type. If a recipe uses 3 cups of flour for every 2 cups of sugar, the ratio is 3 to 2, written 3:2 or 3/2. A rate is a special ratio comparing quantities with different units, such as 120 miles per 2 hours. A unit rate simplifies a rate so the second quantity is 1: 120 miles / 2 hours = 60 miles per 1 hour, or 60 mph. To find a unit rate, divide the first quantity by the second.

Equivalent Ratios: Tables, Tape Diagrams, Double Number Lines

Equivalent ratios name the same relationship. Grade 6 students build ratio tables by scaling both quantities by the same factor:

Each column keeps the 3:2 relationship (6:4 and 9:6 both simplify to 3:2). A tape diagram draws the two quantities as bars partitioned into equal units, three units of flour beside two units of sugar, making the multiplicative structure visible. A double number line places the two quantities on parallel lines with aligned tick marks, so students can slide to any equivalent pair. These models, introduced in grades 6-7, precede the abstract equation and prevent the common error of adding instead of multiplying to scale a ratio.

The Constant of Proportionality and y = kx

Two quantities are proportional when their ratio y : x is constant. That constant is the constant of proportionality, k, and every proportional relationship is captured by:

y = kx, where k = y/x

Here k equals the unit rate. In the recipe, k = sugar/flour = 2/3, so sugar = (2/3) x flour. Grade 7 students learn to test whether a table is proportional by checking that y/x is the same for every row.

Graphing Proportional Relationships and Slope

The graph of y = kx is always a straight line through the origin (0, 0), because when x = 0, y = k(0) = 0. If a line does not pass through the origin, the relationship is not proportional. The unit rate is the slope of that line: slope = rise/run = k. The point (1, k) sits on the line and shows the constant of proportionality directly, because y = k(1) = k.

This links proportional reasoning to similar triangles: any right triangle drawn under the line has legs in the same rise-to-run ratio, so the triangles are similar and the slope is constant everywhere on the line. That is the geometric reason slope does not change along a straight line.

Percent as a Rate Per 100

A percent is a rate per 100. So 45% = 45/100 = 0.45. Because percent is a proportion, the percent proportion is: part/whole = percent/100. To find 45% of 80: (45/100) x 80 = 0.45 x 80 = 36.

Worked Example: Percent Increase and Simple Interest

Percent increase. A jacket costs $60 and its price rises to $75. The increase is 75 - 60 = 15 dollars. Percent increase = (change / original) x 100 = (15 / 60) x 100 = 0.25 x 100 = 25%.

Sales tax and tip. A $60 meal has 8% tax and a 20% tip on the pre-tax amount. Tax = 0.08 x 60 = $4.80. Tip = 0.20 x 60 = $12.00. Total = 60 + 4.80 + 12.00 = $76.80.

Simple interest (I = Prt). Deposit a principal P = $500 at rate r = 4% = 0.04 for t = 3 years. Interest I = P x r x t = 500 x 0.04 x 3 = 20 x 3 = $60. The balance is 500 + 60 = $560.

Percent error. A student measures a length as 52 cm; the true value is 50 cm. Percent error = (|measured - actual| / actual) x 100 = (|52 - 50| / 50) x 100 = (2/50) x 100 = 4%.

Commission. A salesperson earns 6% commission on $2,500 in sales: 0.06 x 2500 = $150.

Discount. A $40 item is 15% off: discount = 0.15 x 40 = $6, so the sale price is 40 - 6 = $34.

Each problem is proportional reasoning at heart: identify the whole, express the percent as a rate per 100, and multiply. Teaching these multistep problems with ratio tables and double number lines helps grades 6-7 students reason rather than memorize disconnected formulas.