Ratios, rates, proportional relationships, and units

Most of these questions are only a sentence or two long. The last line might say:

• “At this rate, how many minutes will it take Dana to pack 130 boxes?”
• “How many feet are equivalent to 222 inches? (1 foot = 12 inches)”
• “What is the density, in grams per cubic centimeter, of the object?”
• “How must the amount of blue paint change to maintain this ratio?”
• “Which of the following is closest to this rate in cubic feet per minute?”

Look for “per,” “at this rate,” “equivalent to” and “proportional to,” and for a conversion fact in parentheses at the end.

Before you calculate anything, write down the unit the answer needs. Say Dana packs 20 boxes in 4 minutes. “How many minutes to pack 130 boxes?” wants 26. “How many boxes per minute?” wants 5.

Desmos works as a calculator here, but it can't tell you which way a conversion goes. On a one-step problem you'll finish faster by hand.

A unit rateThe amount for one unit, like 12 dollars an hour. is the amount for one of something, like dollars per pound or boxes per minute.

An area of 56 square miles has an estimated 1,904 foxes. What is the population density, in foxes per square mile?

For a rate given per dozen or per case, find the amount for one first. Every step after that is a single multiplication or division.

Sometimes two counts change at once. A moving company charged $2,100 for a job where 5 movers each worked 6 hours. At the same rate, what does a job with 3 movers working 4 hours each cost?

1. Count mover-hours.

   $$5\times6=30$$

2. Dollars per mover-hour.

   $$2100\div30=70$$

3. Mover-hours in the new job.

   $$3\times4=12$$

4. The new job costs $840.

   $$12\times70=840$$

Harder questions mix totals with daily averages. Store A sold 4 times as many shirts as Store B in total. Store A averaged 50 shirts a day and Store B averaged 20. Store A's sale lasted how many times as long as Store B's?

You don't know either total, so call Store B's total $T$. The number of days is the total divided by the daily rate.

1. Write each store's total.

   $$\text{A: } 4T \quad \text{B: } T$$

2. Divide each total by its daily rate to get days.

   $$\text{A: } \frac{4T}{50} \quad \text{B: } \frac{T}{20}$$

3. Divide A's days by B's. Dividing by a fraction means multiplying by it flipped.

   $$\frac{4T}{50}\div\frac{T}{20}=\frac{4T}{50}\times\frac{20}{T}$$

4. $T$ is on the top and the bottom, so it cancels.

   $$=\frac{80}{50}=1.6$$

Store A's sale lasted 1.6 times as long.

Dana packs 20 boxes in 4 minutes. At this rate, how many minutes will it take her to pack 130 boxes?

The unit rate is usually the fastest way in.

1. Boxes per minute.

   $$20\div4=5$$

2. Boxes divided by boxes per minute leaves minutes.

   $$130\div5=26$$

You can also set up a proportionTwo ratios set equal to each other, which you can solve for a missing number., which is two ratios set equal. Then cross-multiply, which means multiplying each top by the other side's bottom.

To check in Desmos, type 20/4, press the right arrow, then type =130/x. Desmos draws a vertical line at $x=26$. Without the arrow, the = lands in the bottom of the first fraction and nothing graphs.

A question with no story works the same way. “The ratio $a$ to $b$ is equivalent to the ratio 5 to 8. What is the value of $a$ when $b=56$?” Write $\frac{a}{56}=\frac{5}{8}$ and cross-multiply to get $8a=280$, so $a=35$.

“$y$ is proportional to $x$” means $y=kx$ for one fixed number $k$. If $y=18$ when $x=4$, what is $y$ when $x=10$?

1. Put in the pair you know.

   $$18=k(4)$$

2. Divide both sides by 4.

   $$k=4.5$$

3. Use the same $k$ for the new $x$.

   $$y=4.5(10)=45$$

Density is mass per volume, so it's mass divided by volume. An object has a mass of 234 grams and takes up 18 cubic centimeters. Its density is $234\div18=13$ grams per cubic centimeter.

Harder ones give you the density and only a length. A type of wood has a density of 650 kilograms per cubic meter. A cube of it has edges 0.70 meter long. To the nearest whole number, what is its mass in kilograms?

1. A cube's volume is edge times edge times edge. This is in cubic meters.

   $$0.70^3=0.343$$

2. Mass is density times volume. Cubic meters cancel and kilograms are left.

   $$650\times0.343=222.95$$

3. Round to the nearest kilogram.

   $$\approx223$$

Some questions give the mass and ask for the edge. Divide the mass by the density to get the volume, $222.95\div650=0.343$, then take the cube root. In Desmos, type nthroot, then 3, press the right arrow, and type 0.343. It shows 0.7.

A bag of 70 marbles has red and blue marbles in a ratio of 3 to 7. How many are red?

1. The ratio splits the bag into 10 equal parts.

   $$3+7=10$$

2. Each part holds 7 marbles.

   $$70\div10=7$$

3. Red gets 3 parts, so 21 red.

   $$3\times7=21$$

4. Blue gets 7 parts, so 49 blue.

   $$7\times7=49$$

It works backward too. If red is $\frac{3}{10}$ of the bag, blue is the other $\frac{7}{10}$, so red to blue is 3 to 7.

A paint mix uses 28 ounces of blue for every 16 ounces of white. The painter adds 6 more ounces of white. How must the amount of blue change to maintain this ratio?

1. The new amount of white.

   $$16+6=22$$

2. The multiplier, or how many times bigger the white got.

   $$22\div16=1.375$$

3. Multiply the blue by the same number.

   $$28\times1.375=38.5$$

4. Subtract to find the change.

   $$38.5-28=10.5$$

The blue must increase by 10.5 ounces.

If $\frac{x}{4y}=0.9$ and $\frac{nx}{y}=18$, what is the value of $n$?

You can't find $x$ or $y$ from these two equationsTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson. Both of them contain $\frac{x}{y}$, though, and that's all you need.

1. Start with the first equation.

   $$\frac{x}{4y}=0.9$$

2. Multiply both sides by 4.

   $$\frac{x}{y}=3.6$$

3. Pull $n$ out front in the second equation so $\frac{x}{y}$ shows.

   $$\frac{nx}{y}=n\cdot\frac{x}{y}$$

4. Put in 3.6 for $\frac{x}{y}$.

   $$3.6n=18$$

5. Divide both sides by 3.6.

   $$n=5$$

A conversion fact gives you a fraction worth 1, since its top and bottom are the same amount. Put the unit you're getting rid of on the bottom so it cancels.

How many feet are equivalent to 222 inches? (1 foot = 12 inches)

1. Inches go on the bottom, so they cancel and feet are left.

   $$222\,\cancel{\text{in}}\times\frac{1\text{ ft}}{12\,\cancel{\text{in}}}$$

2. Divide.

   $$=\frac{222}{12}\text{ ft}=18.5\text{ ft}$$

Sometimes the fact comes in an odd form, like (2 meters = 200 centimeters). Cut it down to 1 meter = 100 centimeters before you use it.

A soccer field is 110 yards long. How many inches long is it? (1 yard = 3 feet and 1 foot = 12 inches)

1. Yards cancel, and feet are left.

   $$110\,\cancel{\text{yd}}\times\frac{3\text{ ft}}{1\,\cancel{\text{yd}}}=330\text{ ft}$$

2. Feet cancel, and inches are left.

   $$330\,\cancel{\text{ft}}\times\frac{12\text{ in}}{1\,\cancel{\text{ft}}}=3{,}960\text{ in}$$

An area has two lengths in it, so the conversion has to happen twice. With 1 meter = 100 centimeters:

1. A square meter is 100 centimeters on each side.

   $$1\text{ m}^2=100\text{ cm}\times100\text{ cm}$$

2. So square the factor.

   $$=100^2=10{,}000\text{ cm}^2$$

3. A cube has three lengths, so cube it.

   $$1\text{ m}^3=100^3=1{,}000{,}000\text{ cm}^3$$

How many square meters is 45,000 square centimeters?

If the unit says square or cubic, write the exponentThe small raised number that says how many times to multiply, like the 3 in x cubed.Read the lesson on the factor before you calculate. In Desmos, type 45000/100^2 so you can't type the wrong number of zeros.

On a floor plan, 1 inch represents 8 feet. A wall that's 4.5 inches long on the plan is $4.5\times8=36$ feet long. To go from real feet back to the plan, divide by 8.

A room on the plan is 2 inches by 3 inches. How many square feet is the real room?

1. Turn each plan length into feet.

   $$2\times8=16 \text{ and } 3\times8=24$$

2. Multiply the real lengths to get 384 square feet.

   $$16\times24=384$$

Or start from the room's area on the plan, $2\times3=6$ square inches.

Use one way or the other. Once you've turned both lengths into feet, their product is already in square feet, and multiplying by 64 again would make it 64 times too big.

Miles per hour has a distance unit and a time unit. Convert them one at a time, and write the units on every line.

60 miles per hour is how many feet per second? (1 mile = 5,280 feet and 1 hour = 3,600 seconds)

1. Miles cancel.

   $$\frac{60\,\cancel{\text{mi}}}{1\text{ hr}}\times\frac{5280\text{ ft}}{1\,\cancel{\text{mi}}}=\frac{316{,}800\text{ ft}}{1\text{ hr}}$$

2. Hours cancel. $316{,}800\div3{,}600=88$.

   $$\frac{316{,}800\text{ ft}}{1\,\cancel{\text{hr}}}\times\frac{1\,\cancel{\text{hr}}}{3600\text{ s}}=\frac{88\text{ ft}}{1\text{ s}}$$

The hardest ones change a square or cubic unit and a time unit at once, and give you only a length fact. A tank fills at 4 cubic meters per hour. What is that rate in cubic feet per minute, to the nearest hundredth? (Use 1 meter = 3.28 feet.)

1. The unit is cubic, so cube the length fact.

   $$1\text{ m}^3=3.28^3\text{ ft}^3$$

2. Cubic meters cancel, leaving cubic feet per hour.

   $$4\times3.28^3=141.150208$$

3. Divide by 60 to go from per hour to per minute.

   $$141.150208\div60=2.3525\ldots$$

4. Round once, at the end.

   $$\approx2.35$$

On a grid-inA question where you type your own answer instead of picking one of four.Read the lesson, an answer that doesn't come out even can go in as a fraction. If you get $\frac{20}{3}$, type 20/3. As a decimal it has to fill all five characters, so 6.666 or 6.667. A shorter 6.67 is marked wrong. If Desmos shows a decimal, the small round fraction button on the result row switches it to a fraction.

Leave commas out of big numbers, so the soccer field's 3,960 inches goes in as 3960. The grid-in lesson covers what fits in the box.

Write down the unit each question asks for, and check your answer against it at the end.