6.2 Proportional reasoning & percent applications

Ratios and Proportions

A ratio compares two quantities. The ratio of 3 cups of flour to 2 cups of sugar can be written 3:2, "3 to 2," or as the fraction 3/2. A proportion is a statement that two ratios are equal, such as 3/2 = 9/6. Because the two ratios describe the same relationship, the PERT expects you to solve proportions by cross-multiplying: for a/b = c/d, the cross products are equal, so a . d = b . c.

Worked example 1. If 4 notebooks cost $6, how much do 10 notebooks cost at the same rate? Set up a proportion with notebooks over dollars on both sides: 4/6 = 10/x. Cross-multiply: 4x = 6 . 10 = 60, so x = 15. Ten notebooks cost $15. Keep the units in the same position on both sides (notebooks on top, dollars on the bottom) or the proportion will be set up incorrectly.

Unit Rates

A unit rate is a ratio whose denominator is 1 -- the amount per single unit. To find it, divide the top quantity by the bottom quantity. If a car travels 150 miles on 5 gallons, the unit rate is 150 / 5 = 30 miles per gallon. Speeds work the same way: a runner who covers 12 miles in 2 hours has a unit rate of 12 / 2 = 6 miles per hour. Unit rates make comparison shopping easy: a 12-ounce box for $3.00 costs $0.25 per ounce, while a 20-ounce box for $4.60 costs $0.23 per ounce, so the larger box is the better buy because its per-ounce price is lower.

Percent Basics

Percent means "per hundred," so 37% = 37/100 = 0.37. Two conversions come up constantly:

• Percent to decimal: move the decimal two places left (45% = 0.45).
• Decimal to percent: move the decimal two places right (0.08 = 8%).

The workhorse tool for percent problems is the is/of relationship, often remembered as "is over of equals percent over 100":

part / whole = percent / 100

Worked example 2. What is 20% of 80? Convert the percent to a decimal and multiply: 0.20 x 80 = 16. The word "of" signals multiplication.

Worked example 3. 15 is what percent of 60? Divide the part by the whole: 15 / 60 = 0.25, which is 25%.

Worked example 3b. 9 is 30% of what number? Here the part (9) and the percent (30%) are known, but the whole is missing. Divide the part by the decimal form of the percent: 9 / 0.30 = 30. So 9 is 30% of 30. Notice the pattern: to find the whole, divide the "is" number by the percent; to find the part, multiply.

Percent Change, Discount, and Markup

Percent change measures how much a quantity grew or shrank relative to its original value:

percent change = (new - original) / original x 100

A positive result is an increase; a negative result is a decrease. The most common error is dividing by the new value -- always divide by the original.

Worked example 4. A jacket's price rises from $40 to $50. What is the percent increase? Change = 50 - 40 = 10. Divide by the original: 10 / 40 = 0.25 = 25% increase.

A discount is a percent decrease and a markup is a percent increase. Two reliable methods handle both:

• Two-step: find the amount of the discount, then subtract. An $80 item at 25% off loses 0.25 x 80 = $20, so it costs 80 - 20 = $60.
• One-step multiplier: paying after a 25% discount means paying 100% - 25% = 75%, so 0.75 x 80 = $60. For a 25% markup, multiply by 1.25 instead.

The multiplier method is faster and well worth practicing until it feels automatic.

Worked example 5. A store marks up a $120 bicycle by 15%. What is the selling price? Multiplier = 1 + 0.15 = 1.15. Selling price = 1.15 x 120 = $138.

Simple Interest

Simple interest is interest paid only on the original principal, never on previously earned interest. The formula is I = P . r . t, where P is the principal, r is the annual rate written as a decimal, and t is the time in years.

Worked example 6. You deposit $2,000 at 3% simple interest for 4 years. How much interest do you earn, and what is the balance? Interest: I = 2000 x 0.03 x 4 = $240. Balance = principal + interest = 2000 + 240 = $2,240. If the time were given in months, convert to years first (for example, 6 months = 0.5 year, and 18 months = 1.5 years). Watch the units: the rate and the time must both be annual for the basic formula to work.