Unit 12 · Ratios, Rates and Proportions

A fruit bowl holds 3 apples and 5 oranges. The ratio of apples to oranges is 3 to 5. You can write it as 3:5 or as 3/5. A ratio is a comparison of two amounts by how many of each there are. Order matters. The ratio of apples to oranges is 3:5, but the ratio of oranges to apples is 5:3. Always write the numbers in the same order as the words.

Some ratios compare one part to another part, like apples to oranges. That is a part-to-part ratio. Other ratios compare a part to the whole group. The bowl has 3 + 5 = 8 pieces of fruit, so the ratio of apples to all fruit is 3:8. That is a part-to-whole ratio. A common mistake is to mix these up. In a class of 12 girls and 15 boys, the ratio of girls to boys is 12:15, but the ratio of girls to students is 12:27, because 12 + 15 = 27.

A ratio does not tell you the actual counts. A ratio of 3:5 could mean 3 apples and 5 oranges, or 30 apples and 50 oranges, or 300 and 500. It only tells you how the two amounts compare. That is why people often say a ratio with the words for every: for every 3 apples there are 5 oranges. If you know the ratio and one real count, you can find the other. With 5 oranges for every 3 apples, 15 oranges means 9 apples.

A lemonade recipe uses 2 cups of lemon juice for 6 cups of water, a ratio of 2:6. To make twice as much, double both amounts: 4 cups of juice for 12 cups of water, 4:12. To make half as much, halve both: 1 cup of juice for 3 cups of water, 1:3. All three batches taste the same, because the mix is the same. Ratios that describe the same comparison are called equivalent ratios. You make one by multiplying or dividing both parts by the same number.

The ratio with the smallest whole numbers is the simplest form. To find it, divide both parts by their greatest common factor, the largest number that divides both. For 12:18, the greatest common factor is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3. The simplest form is 2:3. A common mistake is adding the same number to both parts. 2:6 and 3:7 are not equivalent. The first is 1 cup of juice for every 3 cups of water, and the second is much stronger.

To test whether two ratios are equivalent, simplify both and compare. Are 8:20 and 6:15 equivalent? Divide 8:20 by 4 to get 2:5. Divide 6:15 by 3 to get 2:5. They match, so yes. You can also work upward from the simplest form: 2:5 times 4 is 8:20, and 2:5 times 3 is 6:15. Either way, every equivalent ratio comes from the same simplest form.

A basketball league has boys and girls in the ratio 3:2, and 25 players in all. How many are boys? Draw a tape diagram: a strip of 3 equal boxes for boys and a strip of 2 equal boxes for girls. Every box stands for the same number of players. There are 3 + 2 = 5 boxes, and they share 25 players, so each box holds 25 ÷ 5 = 5 players. Boys fill 3 boxes: 3 × 5 = 15. Girls fill 2 boxes: 2 × 5 = 10. Check: 15 + 10 = 25.

Sometimes you know the difference instead of the total. A paint mix uses red and blue in the ratio 5:3, and there are 8 more liters of red than blue. Red has 5 boxes and blue has 3, so red has 5 − 3 = 2 more boxes. Those 2 boxes hold the 8 extra liters, so each box holds 8 ÷ 2 = 4 liters. Red is 5 × 4 = 20 liters and blue is 3 × 4 = 12 liters. Check the difference: 20 − 12 = 8.

The most common mistake is dividing the total by one number from the ratio. For the basketball problem, 25 ÷ 3 gives a decimal that means nothing here. The total is shared by all the boxes, so divide by the sum of the ratio parts. The second mistake is stopping after finding the value of one box. Five players per box is not the answer. Multiply back to get the number of boys.

A bakery sells 5 muffins for $4. A ratio table lists equivalent ratios in rows: 5 muffins for $4, 10 muffins for $8, 15 muffins for $12, 20 muffins for $16. Each row comes from multiplying both numbers in the first row by the same amount. To scale up, multiply; to scale down, divide. The table makes it easy to see what 20 muffins cost, because you can read the answer straight from the row.

You can also add rows. The row 5 muffins for $4 plus the row 10 muffins for $8 gives 15 muffins for $12, and that row is already in the table. Adding works because every row is the same ratio. To find the cost of 35 muffins, notice that 35 = 5 × 7, so the cost is 4 × 7 = $28. Or add rows: 20 muffins plus 15 muffins is 35 muffins, and $16 plus $12 is $28. Both routes agree.

When the number you want is not a neat multiple, go through 1. Five muffins cost $4, so 1 muffin costs 4 ÷ 5 = $0.80. Then 7 muffins cost 7 × 0.80 = $5.60. The most common mistake with a ratio table is changing one column and not the other. If you double the muffins, you must double the dollars. A row that says 10 muffins for $4 is a different, much cheaper bakery.

Jada runs 3 miles every 24 minutes. A double number line shows this with two number lines, one above the other. The top line counts miles: 0, 3, 6, 9. The bottom line counts minutes: 0, 24, 48, 72. Each tick mark on the top line sits directly above its partner on the bottom line. Reading straight down from 6 miles lands on 48 minutes. Both lines start at 0, because 0 miles takes 0 minutes.

To answer a question the lines do not show directly, find the value for 1. Jada takes 24 minutes for 3 miles, so 1 mile takes 24 ÷ 3 = 8 minutes. In 60 minutes she runs 60 ÷ 8 = 7.5 miles. You can see this on the lines too. Sixty minutes sits halfway between 48 and 72. Halfway between their partners, 6 and 9 miles, is 7.5 miles. The picture and the arithmetic agree.

Two mistakes spoil a double number line. The first is uneven spacing. The jumps on each line must be equal, or reading across gives the wrong partner. The second is starting one line at a number other than 0. Both lines must begin at the same 0, because nothing of one amount goes with nothing of the other. A ratio table and a double number line hold the same information, but the line shows the order and the gaps.

A rate is a ratio of two amounts with different units, like miles and hours or dollars and pounds. A car drives 240 miles in 4 hours. To find how far it goes in 1 hour, divide: 240 ÷ 4 = 60 miles per hour. A rate for exactly 1 of the second amount is a unit rate. The word per means for each 1. Sixty miles per hour means 60 miles for each 1 hour.

At the store, 3 pounds of apples cost $5.40. The unit rate is 5.40 ÷ 3 = $1.80 per pound. Every ratio actually has two unit rates. You could also divide the other way: 3 ÷ 5.40 = 0.555…, about 0.56 pounds per dollar. Both are true. Pick the one that answers the question. To compare prices, dollars per pound is the useful one. To see how much a dollar buys, pounds per dollar is.

The common mistake is dividing in the wrong order. For the car, 4 ÷ 240 = 0.0167 hours per mile is a real rate, but it is not miles per hour. The amount before per goes on top and the amount after per is the 1 you divide by. This is exactly what Maya did in the cereal aisle: $3.00 ÷ 12 ounces = $0.25 per ounce. Price first, then divide by the number of ounces.

Back to Maya’s cereal. The small box is 12 ounces for $3.00: 3.00 ÷ 12 = $0.25 per ounce. The large box is 18 ounces for $4.20: 4.20 ÷ 18 = 0.2333…, about $0.23 per ounce. Now both prices are for the same amount, 1 ounce, so you can compare them directly. The large box costs less per ounce, so it is the better deal. Two rates can only be compared when they are in the same units.

The juice showed that bigger is not always cheaper. The 64-ounce bottle for $4.80 is 4.80 ÷ 64 = $0.075 per ounce. The 96-ounce bottle for $7.68 is 7.68 ÷ 96 = $0.08 per ounce. The small bottle wins. A common mistake is to compare the totals, $4.80 and $7.68, and call the cheaper one the better deal. Totals do not tell you the deal, because the bottles hold different amounts.

Rates also compare speeds, and there the bigger number wins. Jo runs 100 meters in 16 seconds: 100 ÷ 16 = 6.25 meters per second. Kim runs 200 meters in 34 seconds: 200 ÷ 34 = 5.88 meters per second. Jo is faster, even though Kim ran farther. So read what the rate measures before you decide. For a price per ounce, smaller is better. For meters per second, bigger is better.

A tutor earns $45 for 3 hours of work. The unit rate is 45 ÷ 3 = $15 per hour. Now you can predict pay for any number of hours. For 7 hours: 7 × 15 = $105. You can also go backward. If the tutor earned $120, divide by the unit rate: 120 ÷ 15 = 8 hours. The unit rate is the bridge between the two amounts in both directions.

Speed works the same way. Driving from Chicago to Springfield is a trip of about 200 miles. At 60 miles per hour, the time is 200 ÷ 60 = 3.33… hours, about 3 hours and 20 minutes, because 0.33 of an hour is 20 minutes. If you know the time instead, multiply. Two and a half hours at 60 miles per hour covers 2.5 × 60 = 150 miles.

The common mistake is multiplying when you should divide, or the reverse. Ask yourself: am I finding the total, or the number of units? Finding a total from a number of units means multiply. Finding the number of units from a total means divide. The units on the numbers help: $15 per hour × 7 hours = $105, and the hours cancel. $120 ÷ $15 per hour = 8 hours, and the dollars cancel.

There are 12 inches in 1 foot. That is a rate, 12 inches per foot, and using it is a unit conversion. To change 5.5 feet to inches, multiply: 5.5 × 12 = 66 inches. To change 90 inches to feet, divide: 90 ÷ 12 = 7.5 feet. The number 12 is the conversion factor, the unit rate between the two units. Every conversion works the same way: multiply to get the smaller unit, divide to get the larger one.

The metric system uses conversion factors that are powers of 10. There are 1,000 meters in a kilometer, 1,000 grams in a kilogram and 100 centimeters in a meter. So 3.2 kilometers is 3.2 × 1,000 = 3,200 meters, and 450 grams is 450 ÷ 1,000 = 0.45 kilograms. The ratio of meters to kilometers is 1,000:1, and every metric conversion is a ratio like that.

Some conversions take two steps. A speed of 30 miles per hour in miles per minute: 1 hour is 60 minutes, so 30 ÷ 60 = 0.5 miles per minute. A well-known result: 60 miles per hour is 88 feet per second, because 60 × 5,280 feet ÷ 3,600 seconds = 88. The common mistake is multiplying when you should divide. Check with sense: converting to a smaller unit gives a bigger number, since it takes more inches than feet to measure the same wall.

A rice recipe for 4 servings uses 3 cups of rice. You need 10 servings. Find the rate per serving: 3 ÷ 4 = 0.75 cups per serving. Then 10 × 0.75 = 7.5 cups. A ratio table gets there too: 4 servings use 3 cups, 8 servings use 6 cups, 2 servings use 1.5 cups, so 10 servings use 6 + 1.5 = 7.5 cups. Both methods agree.

A map’s scale is a ratio between a length on the map and the real distance. Suppose a map says 1 inch stands for 25 miles. Two cities 3.5 inches apart on the map are 3.5 × 25 = 87.5 miles apart. Going backward, a real distance of 200 miles, about the drive from Chicago to Springfield, is 200 ÷ 25 = 8 inches on that map. The scale is a unit rate: 25 miles per inch.

Speed is a rate of distance per time. A cyclist rides 18 miles in 1.5 hours, so her speed is 18 ÷ 1.5 = 12 miles per hour. A 30-mile ride at that speed takes 30 ÷ 12 = 2.5 hours. The common mistake is mixing minutes and hours. If the ride took 90 minutes, convert first: 90 minutes is 1.5 hours. Dividing 18 by 90 gives miles per minute, a different rate, and comparing it with miles per hour is meaningless.

A bike rental shop charges $6 per hour. One hour costs $6, 2 hours cost $12, 3 hours cost $18 and 5 hours cost $30. Divide the cost by the hours in each row: 6 ÷ 1 = 6, 12 ÷ 2 = 6, 18 ÷ 3 = 6, 30 ÷ 5 = 6. The answer is always 6. When two amounts always have the same ratio like this, they are in a proportional relationship. The number that stays the same, here 6, is a constant.

A second shop charges a $10 fee plus $4 per hour. One hour costs $14, 2 hours cost $18 and 3 hours cost $22. Divide again: 14 ÷ 1 = 14, but 18 ÷ 2 = 9, and 22 ÷ 3 = 7.33…. The ratio changes, so this relationship is not proportional. There is another giveaway. In a proportional relationship, 0 of one amount goes with 0 of the other. Renting for 0 hours at the first shop costs $0, but at the second shop it still costs the $10 fee.

The test is simple: divide y by x in every row and see whether you always get the same number. A common mistake is checking only whether y goes up by the same amount each time. The second shop’s costs go up by 4 each hour, 14, 18, 22, and that steady climb looks proportional, but it is not, because the ratio y ÷ x keeps changing. Equal steps are not enough. The ratio must be equal in every row.

At the $6-per-hour bike shop, the cost y for x hours is y = 6 × x, written y = 6x. The number 6 is the constant of proportionality, usually called k. It is the unit rate: the cost for 1 hour. Every proportional relationship has an equation of the form y = kx. To find k from any row of a table, divide y by x. If a table shows x = 4 and y = 22, then k = 22 ÷ 4 = 5.5, and the equation is y = 5.5x.

Once you have the equation, you can find any pair. With y = 5.5x, when x = 12, y = 5.5 × 12 = 66. Going backward, when y = 99, solve 99 = 5.5x by dividing: x = 99 ÷ 5.5 = 18. Check: 5.5 × 18 = 99. The equation replaces a whole table.

Three T-shirts cost $27. Find the unit rate: 27 ÷ 3 = 9, so k = 9 and y = 9x. Eleven shirts cost 9 × 11 = $99. The common mistake is writing y = x + 9, adding instead of multiplying. Test it with 2 shirts. The table says 2 shirts cost 9 × 2 = $18. The wrong equation gives 2 + 9 = 11, which is not what the store charges.

Plot the bike shop’s table as points: (0, 0), (1, 6), (2, 12), (3, 18). The points line up on a straight line, and that line passes through the origin, the point (0, 0) where the axes cross. Every proportional relationship makes a graph like this: a straight line through the origin. The point where x is 1, here (1, 6), shows the unit rate. Its y-value is the constant of proportionality.

The $10-fee shop makes a straight line too, but it crosses the y-axis at 10, not at 0, so it is not proportional. A curve is not proportional either, even if it starts at the origin. Reading a graph runs backward as well. If a cost graph goes through the point (4, 30), then 4 hours cost $30, and k = 30 ÷ 4 = 7.5. From one point you can write the whole equation, y = 7.5x.

Two proportional graphs on the same axes can be compared by steepness. The steeper line has the bigger k. A walker’s distance line passes through (2, 6), so she walks 6 ÷ 2 = 3 miles per hour. A biker’s line passes through (2, 24), so he rides 24 ÷ 2 = 12 miles per hour. The biker’s line climbs faster. The common mistake is reading a point in the wrong order. The first number is always x, and the second is always y.

A percent is a ratio whose second number is 100. The word comes from per hundred. Saying 45% means 45 out of every 100, the ratio 45:100, the fraction 45/100 and the decimal 0.45 all at once. Percents make ratios easy to compare because every one is out of the same 100. In a class of 25 students, 20 passed a test. Scale the ratio 20:25 up to 100: multiply both by 4 to get 80:100. So 80% passed.

You will move between fractions, decimals and percents constantly. Fraction to percent: divide, then multiply by 100. For 3/8, 3 ÷ 8 = 0.375, and 0.375 × 100 = 37.5%. Percent to decimal: divide by 100, which moves the decimal point two places left. So 6% is 0.06, not 0.6. That is the most common slip, and it makes every answer ten times too big. Decimal to percent: multiply by 100. So 1.25 is 125%.

Some benchmark percents are worth knowing by heart because they come up so often. 50% is 1/2, 25% is 1/4, 20% is 1/5, 10% is 1/10 and 1% is 1/100. 100% is the whole thing. A percent can be larger than 100: if a plant grows from 20 cm to 50 cm, its new height is 250% of the old one, because 50 ÷ 20 = 2.5. Knowing the benchmarks lets you build others: 15% is 10% plus 5%, and 5% is half of 10%.

The store took 30% off an $80 jacket. To find 30% of 80, turn the percent into a decimal and multiply: 0.30 × 80 = 24. So the discount was $24. Benchmarks give the same answer another way: 10% of 80 is 8, so 30% is 3 × 8 = 24. Or use the fraction: 30/100 × 80 = 2,400/100 = 24. The percent is the ratio, the whole is the number you start with and the part is the answer.

The same equation, part = percent × whole, answers three kinds of questions. To find the part: 30% of 80 is 0.30 × 80 = 24. To find the whole: 24 is 30% of what number? Divide: 24 ÷ 0.30 = 80. To find the percent: 24 is what percent of 80? Divide the part by the whole: 24 ÷ 80 = 0.30, which is 30%. Decide which of the three numbers is missing, then multiply or divide to get it.

For mental math, break the percent into benchmarks. For 15% of 60, take 10% of 60, which is 6, and 5% of 60, which is half of that, 3. Add: 6 + 3 = 9. The common mistake is multiplying by the percent number instead of the decimal. 30 × 80 = 2,400 is far too big for 30% of 80. Sanity check every answer: 30% is less than half, so 30% of 80 must be less than 40.

Rent goes from $800 a month to $880. The change is 880 − 800 = $80. To describe that change as a percent, divide it by the original amount: 80 ÷ 800 = 0.10, which is 10%. That is a percent increase of 10%. For a percent decrease, the same rule applies. A price drops from $50 to $40. The change is 10, and 10 ÷ 50 = 0.20, so the price fell 20%.

The common mistake is dividing by the new amount. For the price drop from 50 to 40, dividing 10 by 40 gives 25%, which is wrong. The percent change always compares the change to where you started. To apply a percent change, find the change and add or subtract it. A 25% increase on $60: 0.25 × 60 = 15, and 60 + 15 = $75. A faster way is one multiplication: a 25% increase means paying 125%, so 1.25 × 60 = 75. A 25% decrease means paying 75%, so 0.75 × 60 = 45.

An increase and a decrease of the same percent do not cancel out. Start with 100, increase by 20% to 120, then decrease by 20%. Twenty percent of 120 is 24, so the result is 96, not 100. The decrease is bigger in dollars because it was taken from a bigger number. This is the same reason Devon’s two discounts did not add up: each percent is figured from a different amount.

A discount is a percent taken off a price. Devon’s jacket at $80 with 30% off: 0.30 × 80 = 24 off, so the sale price is 80 − 24 = $56. The one-step way is to multiply by what is left: 100% − 30% = 70%, and 0.70 × 80 = 56. Coupons, sales and student discounts all work like this.

Sales tax is a percent added to a price. Tax rates differ from city to city, so suppose a city’s rate is 10%. A $40 restaurant meal has 0.10 × 40 = $4 of tax, so the bill is $44. A tip is another percent added, usually figured from the price of the food before tax. An 18% tip on the $40 meal is 0.18 × 40 = $7.20. The total paid is 40 + 4 + 7.20 = $51.20. To find a 20% tip fast, take 10% and double it: 10% of 40 is 4, so 20% is 8.

When a discount and a tax both apply, take the discount first, then tax the sale price. The $56 jacket with 8% tax: 0.08 × 56 = 4.48, so the total is 60.48. Or 1.08 × 56 = 60.48. The common mistake is taxing the original $80, which overcharges. And as Devon learned, two discounts in a row are figured one after the other: 80 × 0.70 = 56, then 56 × 0.80 = 44.80. Never add the percents.

When you put money in a savings account, the bank pays you interest, a percent of your money, for the time it stays there. With simple interest, the percent is figured from the starting amount, called the principal, every year. The rule is interest = principal × rate × time. Put $500 in an account at 4% per year (an example rate) for 3 years: 500 × 0.04 × 3 = $60. The account then holds 500 + 60 = $560.

Loans work the same way, but you pay the interest. Borrow $1,200 at 6% per year for 2 years: 1,200 × 0.06 × 2 = $144 of interest. You pay back 1,200 + 144 = $1,344. Two common mistakes: forgetting to multiply by the time, and using 6 instead of 0.06. The second one gives 1,200 × 6 × 2 = 14,400, which is more than ten times the loan. A sanity check catches it: 6% of anything is a small slice.

Percent error measures how far off a guess or a measurement is. You guess a jar holds 45 jelly beans, and the true count is 50. The error is 50 − 45 = 5. Divide by the true value: 5 ÷ 50 = 0.10, so the percent error is 10%. A ruler reading of 20 cm for a board that is really 25 cm long has an error of 5, and 5 ÷ 25 = 20% error. Always divide by the true value, not the guess, just as percent change always divides by the original.

A floor plan of an apartment uses the scale 1 inch to 4 feet. Every inch on the paper stands for 4 feet in the real rooms. A bedroom that is 3 inches long on the plan is 3 × 4 = 12 feet long. A drawing like this is a scale drawing, and the number that turns paper lengths into real lengths, here 4 feet per inch, is the scale factor. It is a constant of proportionality: real length = 4 × drawn length.

Go backward by dividing. A living room 20 feet wide is 20 ÷ 4 = 5 inches wide on the plan. If you redraw the plan at 1 inch to 2 feet, every length on the paper doubles, because each inch now stands for less. The 12-foot bedroom becomes 12 ÷ 2 = 6 inches long. The rooms did not change; only the scale did.

Area does not scale by the same number as length. The bedroom is 3 inches by 2.5 inches on the plan, an area of 7.5 square inches. In real life it is 12 feet by 10 feet, an area of 120 square feet. The lengths grew by 4, but the area grew by 4 × 4 = 16, and 7.5 × 16 = 120. The common mistake is multiplying the area by the scale factor once. When lengths scale by k, areas scale by k × k.

Every percent change has a multiplier: the decimal you multiply by to get the new amount. Thirty percent off means keeping 70%, a multiplier of 0.70. Twenty percent off means a multiplier of 0.80. Devon’s jacket: 80 × 0.70 = 56, then 56 × 0.80 = 44.80. Multiply the two multipliers together: 0.70 × 0.80 = 0.56. Devon paid 56% of the original price, so his total discount was 100% − 56% = 44%, not 50%.

Successive increases behave the same way. A $200 phone plan rises 10% one year and 10% the next. The multiplier for a 10% increase is 1.10. So 200 × 1.10 = 220, then 220 × 1.10 = 242. The two increases together are 1.10 × 1.10 = 1.21, a 21% rise, not 20%. The second increase was figured on the bigger $220, so it added more dollars than the first.

The order of the multipliers does not matter, because 0.70 × 0.80 and 0.80 × 0.70 are the same number. That is why a discount followed by tax gives the same total as tax followed by the same discount. Devon’s $44.80 jacket with 8% sales tax: 44.80 × 1.08 = 48.38, rounded to the cent. The only mistake to avoid is adding the percents. Two discounts of 30% and 20% are never 50% off, and two 10% raises are always a bit more than 20%.