Ratios, Rates & Proportions

Study Sheet

Ratios, Rates & Proportions

Writing and simplifying ratios, equivalent ratios, unit rates and best buys, solving proportions, scale drawings, and similar figures

What a Ratio Is

Concept
The Rule

A ratio compares two quantities by division — it tells you how much of one thing there is for each amount of another. If a bowl holds 33 apples and 55 oranges, the ratio of apples to oranges is “33 to 55.”

There are three ways to write the same ratio:

  • with the word to:   33 to 55
  • with a colon:   3:53:5
  • as a fraction:   35\frac{3}{5}

Order matters! The ratio of apples to oranges (3:53:5) is different from oranges to apples (5:35:3). To simplify a ratio, divide both numbers by their greatest common factor, just like reducing a fraction.

A proportion is two equal ratios, ab=cd\tfrac ab = \tfrac cd, and cross-multiplying turns it into ad=bcad = bc, an equation you can solve. Similar figures, scale drawings, unit prices and speeds are all proportions. Keep the same quantity on top in both fractions and the setup takes care of itself.

Simplifying a ratio is finding the smallest repeating group. The 1212 cars and 88 trucks can be sorted into 44 identical groups, each with 33 cars and 22 trucks; the ratio 3:23:2 describes one group, and the lot is just four copies of it. That is why dividing both parts by the same number (here the GCF, 44) never changes the ratio.

Example
Worked Example: three ways & simplifying
4 groups of (3 cars, 2 trucks): 12 : 8 = 3 : 2
Every group has 33 cars (squares) for 22 trucks (circles).

A parking lot has 1212 cars and 88 trucks. Write the ratio of cars to trucks.

  • Three ways: 1212 to 88,   12:812:8,   128\frac{12}{8}.
  • Both numbers share a factor of 44:   12÷4=3\;12 \div 4 = 3 and 8÷4=28 \div 4 = 2.
  • Simplest form: 3:2\mathbf{3:2} — for every 33 cars there are 22 trucks.
Side note
Where it goes wrong

Writing the ratio of cars to trucks as 8:128:12 because trucks were mentioned first somewhere. The order of the words "cars to trucks" fixes the order of the numbers: 12:8=3:212:8 = 3:2.

Example
Worked Example: comparing with a total

In a class of 1010 boys and 1515 girls, the ratio of boys to girls is 10:15=2:310:15 = \mathbf{2:3}. The ratio of boys to all students is 10:25=2:510:25 = \mathbf{2:5}, because the class has 10+15=2510 + 15 = 25 students in all.

Tip

Tip: A ratio in simplest form has no common factors left, but you never turn it into a single number — keep both parts. Write 2:32:3, not “0.670.67.” Always state the order in words so you don't flip it.

Try it
Try it: part to part and part to whole

A playlist has 1414 pop songs and 2121 rock songs. Give the ratio of pop to rock, and of rock to all songs, in simplest form.

Show the answer

Pop to rock is 14:2114:21; both share 77, so 2:32:3. There are 14+21=3514 + 21 = 35 songs, so rock to all is 21:35=3:521:35 = 3:5.

Equivalent Ratios & Ratio Tables

Concept
The Rule

Two ratios are equivalent if they simplify to the same ratio. You build an equivalent ratio by multiplying or dividing both parts by the same number — exactly how you make equivalent fractions.

23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

A ratio table lists several equivalent ratios in a row, which makes patterns easy to see and missing values easy to fill in.

A ratio table lists batches of the same recipe. One batch is 22 juice to 55 soda; two batches double both amounts, three batches triple them, and every column tastes the same. Adding the same number to both parts (from 2:52:5 to 3:63:6) is not another batch; it changes the flavor, which is why only multiplying and dividing are allowed.

Example
Worked Example: filling a ratio table
juicesoda00254106158201025
Each tick is one more batch of 22 juice to 55 soda.

A punch recipe mixes juice to soda in the ratio 2:52:5. Complete the table.

juice (cups)24610soda (cups)5101525\begin{array}{c|cccc} \text{juice (cups)} & 2 & 4 & 6 & 10 \\ \hline \text{soda (cups)} & 5 & 10 & 15 & 25 \end{array}

Each column keeps the ratio 2:52:5: multiply both parts of 2:52:5 by 22 to get 4:104:10, by 33 to get 6:156:15, and by 55 to get 10:2510:25.

Side note
Where it goes wrong

Filling a ratio table by adding: from 2:52:5 to 4:74:7 ("add 22 to each"). Check the batch: 4:74:7 does not simplify to 2:52:5. Doubling the juice must double the soda: 4:104:10.

Example
Worked Example: are they equivalent?

Is 6:106:10 equivalent to 9:159:15? Simplify each: 6:10=3:56:10 = 3:5 and 9:15=3:59:15 = 3:5. They match, so yes, they are equivalent.

Tip

Tip: You must multiply (or divide) both numbers by the same amount. Changing only one part, or adding instead of multiplying, breaks the ratio.

Try it
Try it: complete the table

Paint is mixed blue to white in the ratio 3:43:4. How much white goes with 1212 cans of blue? How much blue goes with 1010 cans of white?

Show the answer

1212 is 3×43 \times 4, so multiply both parts by 44: 12:1612:16, so 1616 cans of white. 1010 is 4×2.54 \times 2.5, so multiply by 2.52.5: 7.5:107.5:10, so 7.57.5 cans of blue.

Rates & Unit Rates

Concept
The Rule

A rate is a ratio that compares two quantities with different units, such as miles and hours, or dollars and pounds. A unit rate is a rate with a denominator of 1: how much of the first quantity for exactly one of the second, like miles per hour or dollars per pound. To find a unit rate, divide the first quantity by the second.

A unit price is a unit rate for cost (dollars per item or per unit). Comparing unit prices tells you the best buy — the lower unit price is the better deal.

A unit rate is the ratio scaled down to exactly one of the second quantity, and that makes comparisons fair. Twelve dollars for six bottles and twenty-five dollars for ten cannot be compared directly, because the packs are different sizes; dividing each by its number of bottles puts both on the same footing: dollars for one bottle.

Example
Worked Example: finding a unit rate
mileshours0050110021503
150150 miles in 33 hours is 5050 miles for each 11 hour.

A car travels 150150 miles in 33 hours. The unit rate is

150 miles3 hours=50 miles1 hour=50 miles per hour.\frac{150 \text{ miles}}{3 \text{ hours}} = \frac{50 \text{ miles}}{1 \text{ hour}} = \mathbf{50 \text{ miles per hour}}.

Divide 150÷3=50150 \div 3 = 50. The word per means “for each one.”

Side note
Where it goes wrong

Dividing the wrong way: 3÷150=0.023 \div 150 = 0.02 "miles per hour." That number is hours per mile. The unit named first (miles) goes on top: 150÷3=50150 \div 3 = 50 miles per hour.

Example
Worked Example: best buy by unit price

A 66-pack of juice costs $1212 and a 1010-pack costs $2525. Which is the better buy?

  • 66-pack:   $12÷6=$2.00\;\$12 \div 6 = \$2.00 per bottle.
  • 1010-pack:   $25÷10=$2.50\;\$25 \div 10 = \$2.50 per bottle.

The 66-pack is cheaper per bottle, so the 66-pack is the better buy.

Tip

Tip: Keep the units consistent before you compare. If one price is per gram and another is per kilogram, convert so both use the same unit first, then compare.

Try it
Try it: best buy

A 1616-ounce jar of peanut butter costs $3.20\$3.20 and a 2828-ounce jar costs $5.32\$5.32. Which is the better buy?

Show the answer

Unit prices: 3.20÷16=0.203.20 \div 16 = 0.20 dollars per ounce and 5.32÷28=0.195.32 \div 28 = 0.19 dollars per ounce. The 2828-ounce jar costs 11 cent less per ounce, so it is the better buy.

Writing & Solving Proportions

Concept
The Rule

A proportion is an equation stating that two ratios are equal, like 23=812\frac{2}{3} = \frac{8}{12}. When one part is unknown, we solve for it using cross-multiplication: in a true proportion, the two diagonal products are equal.

ab=cd⟹a×d=b×c.\frac{a}{b} = \frac{c}{d} \quad\Longrightarrow\quad a \times d = b \times c.

Multiply across the diagonals, set the products equal, then divide to isolate the unknown.

Cross-multiplying is clearing denominators. Starting from ab=cd\tfrac{a}{b} = \tfrac{c}{d} and multiplying both sides by bdbd gives ad=bcad = bc; both sides were multiplied by the same nonzero number, so the equation is still true. That is also why it only applies to an equation of two fractions, not to a sum like ab+cd\tfrac{a}{b} + \tfrac{c}{d}.

Example
Worked Example: solving with cross-multiplication

Solve x4=68\dfrac{x}{4} = \dfrac{6}{8}.

  • Cross-multiply: 8×x=4×68 \times x = 4 \times 6, so 8x=248x = 24.
  • Divide both sides by 88: x=24÷8=3x = 24 \div 8 = \mathbf{3}.

Check: 34=68\frac{3}{4} = \frac{6}{8}? Yes, both simplify to 34\frac{3}{4}.

Side note
Where it goes wrong

Cross-multiplying and then dividing by the wrong number: from 8x=248x = 24 writing x=8÷24=13x = 8 \div 24 = \tfrac13. The 88 multiplies xx, so divide 2424 by 88: x=3x = 3. Check in the original: 34=68\tfrac34 = \tfrac68.

Example
Worked Example: an answer that is not a whole number

Solve 58=x20\dfrac{5}{8} = \dfrac{x}{20}. Cross-multiply: 8x=5×20=1008x = 5 \times 20 = 100. Divide: x=100÷8=12.5x = 100 \div 8 = \mathbf{12.5}. Ratios don't always give whole-number answers, and that is perfectly fine.

Tip

Tip: Set up a proportion so matching units line up — keep the same kind of quantity in the same position (top with top, bottom with bottom). Cross-multiply to solve.

Try it
Try it: solve

Solve 7x=2136\dfrac{7}{x} = \dfrac{21}{36} and x+25=915\dfrac{x + 2}{5} = \dfrac{9}{15}.

Show the answer

First: 21x=7×36=25221x = 7 \times 36 = 252, so x=12x = 12. Check: 712=2136\tfrac{7}{12} = \tfrac{21}{36}, both equal to 712\tfrac{7}{12}. Second: 15(x+2)=4515(x + 2) = 45, so x+2=3x + 2 = 3 and x=1x = 1. Check: 35=915\tfrac35 = \tfrac{9}{15}.

Using Proportions to Solve Word Problems

Concept
The Rule

Many everyday problems are really proportions in disguise: recipes, prices, speeds, and mixtures. To solve one:

  • Write a ratio from the known facts, labeling the units.
  • Set it equal to a second ratio in the same order, using a letter for the unknown.
  • Cross-multiply and solve.

A proportion is the right model exactly when the two quantities scale together: twice the cookies need twice the flour. When they do, the ratio stays fixed, so the known pair and the unknown pair are equal fractions. Before setting one up, ask whether doubling one quantity really doubles the other; for things like a flat fee plus a rate, it does not.

Example
Worked Example: a recipe problem
cupscookies0016212318424530
22 cups for 1212 cookies, so 11 cup per 66; 3030 cookies need 55 cups.

A recipe uses 22 cups of flour to make 1212 cookies. How much flour makes 3030 cookies?

2 cups12 cookies=x cups30 cookies\frac{2 \text{ cups}}{12 \text{ cookies}} = \frac{x \text{ cups}}{30 \text{ cookies}}

Cross-multiply: 12x=2×30=6012x = 2 \times 30 = 60, so x=60÷12=5x = 60 \div 12 = \mathbf{5} cups. Notice cups stayed on top and cookies on the bottom on both sides.

Side note
Where it goes wrong

Using a proportion when the quantities do not scale: a taxi costing $3\$3 plus $2\$2 per mile charges $13\$13 for 55 miles, but 1010 miles cost $23\$23, not $26\$26. The flat fee is paid once, so the cost is not proportional to distance.

Example
Worked Example: a price problem

If 44 tickets cost $2020, how much do 77 tickets cost? Set up 420=7x\frac{4}{20} = \frac{7}{x} (tickets over dollars). Cross-multiply: 4x=20×7=1404x = 20 \times 7 = 140, so x=140÷4=$35x = 140 \div 4 = \mathbf{\$35}.

Tip

Tip: Always write the units next to your numbers as you set up the proportion. Matching units in matching spots is the surest way to avoid flipping the ratio.

Try it
Try it: set up and solve

A printer prints 4545 pages in 33 minutes. How long does it take to print 120120 pages?

Show the answer

Pages and minutes scale together. 45 pages3 min=120 pagesx min\tfrac{45 \text{ pages}}{3 \text{ min}} = \tfrac{120 \text{ pages}}{x \text{ min}}, so 45x=36045x = 360 and x=8x = 8 minutes. (Unit rate check: 1515 pages per minute, and 120÷15=8120 \div 15 = 8.)

Scale Drawings, Maps & Models

Concept
The Rule

A scale tells how a drawing, map, or model relates to the real object. It is written as a ratio, such as “11 inch =20= 20 miles” or 1:181:18. Every scale problem is a proportion:

scale drawing sizereal size=drawing measurementactual measurement.\frac{\text{scale drawing size}}{\text{real size}} = \frac{\text{drawing measurement}}{\text{actual measurement}}.

Set up the scale ratio, then cross-multiply to find the missing distance — either the real distance from a drawing, or the drawing size from a real distance.

A scale drawing is a similar copy of the real object, so every length on it is multiplied by the same scale factor. That single factor converts any distance, which is why one scale statement ("11 inch =20= 20 miles") answers questions about every pair of cities on the map.

Example
Worked Example: reading a map
AshbyBramley4 in on the map0 mi20 mi40 mi60 mi80 miscale bar: each 1 in = 20 mi
Four scale-bar inches: 4×20=804 \times 20 = 80 miles.

On a map, 11 inch represents 2020 miles. Two cities are 44 inches apart on the map. How far apart are they really?

1 in20 mi=4 inx mi  ⟹  1⋅x=20×4=80.\frac{1 \text{ in}}{20 \text{ mi}} = \frac{4 \text{ in}}{x \text{ mi}} \;\Longrightarrow\; 1 \cdot x = 20 \times 4 = 80.

The cities are 80\mathbf{80} miles apart.

Side note
Where it goes wrong

Scaling area with the length factor: a 1:181:18 model car with roof area 4040 cm2^2 does not have a real roof of 40×18=72040 \times 18 = 720 cm2^2. Areas scale by the square of the factor: 40×182=12,96040 \times 18^2 = 12{,}960 cm2^2.

Example
Worked Example: a scale model

A model car is built at scale 1:181:18, meaning 11 cm on the model equals 1818 cm on the real car. If the model is 2525 cm long, the real car is 25×18=45025 \times 18 = \mathbf{450} cm long (that is 4.54.5 meters).

Tip

Tip: Keep units consistent and keep the scale ratio in the same order on both sides (drawing on top, real on bottom). A quick sanity check: real distances should come out much larger than the drawing.

Try it
Try it: a floor plan

On a floor plan, 11 cm represents 0.50.5 m. A room measures 88 cm by 6.56.5 cm on the plan. What are its real dimensions?

Show the answer

Multiply each plan length by 0.50.5 m per cm: 8×0.5=48 \times 0.5 = 4 m and 6.5×0.5=3.256.5 \times 0.5 = 3.25 m. The room is 44 m by 3.253.25 m.

Similar Figures & Missing Sides

Concept
The Rule

Two figures are similar when they have the same shape but not necessarily the same size — one is a scaled copy of the other. In similar figures, corresponding sides are proportional: they all share the same scale factor. To find a missing side, set up a proportion of corresponding sides and solve.

side of figure 1matching side of figure 2=another side of figure 1its matching side of figure 2.\frac{\text{side of figure 1}}{\text{matching side of figure 2}} = \frac{\text{another side of figure 1}}{\text{its matching side of figure 2}}.

Similar figures are scaled copies: one scale factor multiplies every length. Here the short side goes from 33 to 66, a factor of 22, so the long side must go from 44 to 88. The proportion 36=4x\tfrac36 = \tfrac4x is the same statement written as equal ratios, and it works for any matching pair of sides.

Example
Worked Example: missing side of a similar rectangle
43x = 86
Every side doubles: 3→63 \to 6 and 4→84 \to 8.

A small rectangle is 33 by 44. A similar large rectangle has its short side equal to 66 (matching the 33). Find the long side xx (matching the 44).

36=4x  ⟹  3x=6×4=24  ⟹  x=8.\frac{3}{6} = \frac{4}{x} \;\Longrightarrow\; 3x = 6 \times 4 = 24 \;\Longrightarrow\; x = \mathbf{8}.

The scale factor is 22 (since 6=3×26 = 3 \times 2), and indeed 4×2=84 \times 2 = 8.

Side note
Where it goes wrong

Matching sides out of order: 36=x4\tfrac36 = \tfrac{x}{4}, giving x=2x = 2. That pairs the small rectangle's long side with the large one's unknown on the wrong side of the fraction. Keep each fraction "small figure over large figure": 36=4x\tfrac36 = \tfrac4x.

Example
Worked Example: shadows and similar triangles

At the same time of day, a 55-ft person casts a 66-ft shadow while a tree casts a 2424-ft shadow. The two right triangles are similar, so

5 ft6 ft=h24 ft  ⟹  6h=5×24=120  ⟹  h=20 ft.\frac{5 \text{ ft}}{6 \text{ ft}} = \frac{h}{24 \text{ ft}} \;\Longrightarrow\; 6h = 5 \times 24 = 120 \;\Longrightarrow\; h = \mathbf{20 \text{ ft}}.

The tree is 2020 feet tall.

Tip

Tip: Match corresponding sides carefully — pair the shortest with the shortest, and so on. Every pair of corresponding sides in similar figures shares the very same scale factor.

Try it
Try it: similar triangles

Triangle ABCABC has sides 55, 77, 99. A similar triangle has shortest side 1515. Find its other two sides.

Show the answer

The shortest sides correspond, so the scale factor is 15÷5=315 \div 5 = 3. The other sides are 7×3=217 \times 3 = 21 and 9×3=279 \times 3 = 27.

Going Deeper: Advanced Ratio & Proportion Ideas

Concept
Direct vs. Inverse Variation

When two quantities stay in a constant ratio, we say yy varies directly with xx:

y=kxequivalentlyyx=k  (constant).y = kx \qquad\text{equivalently}\qquad \frac{y}{x} = k \;(\text{constant}).

Here kk is the constant of proportionality. Double xx and yy doubles too — the graph is a straight line through the origin.

When instead the two quantities have a constant product, yy varies inversely with xx:

y=kxequivalentlyx⋅y=k  (constant).y = \frac{k}{x} \qquad\text{equivalently}\qquad x \cdot y = k \;(\text{constant}).

Now doubling xx halves yy. More workers means less time; faster speed means fewer hours for the same trip. The tell-tale sign of inverse variation is that the product stays fixed while the ratio does not.

Example
Worked Example: spotting and using inverse variation
pd246810121416182468101214(6, 8)(16, 3)
Painters pp times days dd is 4848 at every point of the curve.

It takes 66 painters 88 days to paint a mural. Working at the same pace, how long will 1616 painters take? More painters means less time, so this is inverse variation: painters ×\times days is constant.

  • Find the constant: k=6×8=48k = 6 \times 8 = 48 (painter-days of work).
  • Solve for the new time: 16×d=4816 \times d = 48, so d=48÷16=3d = 48 \div 16 = \mathbf{3} days.

Warning: a beginner might set up a direct proportion 68=16d\frac{6}{8} = \frac{16}{d} and get the wrong answer. Ask first: as one quantity grows, does the other grow (direct) or shrink (inverse)?

Concept
Three-Term Ratios a:b:ca:b:c

A ratio can compare three or more quantities at once, such as 2:3:42:3:4. Think of it as splitting a whole into parts: 2:3:42:3:4 uses 2+3+4=92+3+4 = 9 equal parts. To share an amount in a given ratio, divide by the total number of parts to find one part, then multiply.

You can also combine two separate two-term ratios into one. If A:B=2:3A:B = 2:3 and B:C=6:5B:C = 6:5, first rescale so the shared term BB matches. Multiply the first ratio by 22 to get A:B=4:6A:B = 4:6; now BB is 66 in both, giving A:B:C=4:6:5A:B:C = 4:6:5.

Example
Worked Example: sharing money in a ratio
$20$20$20$20$20$20$20$20$20$40$60$80
99 equal parts of $20\$20, split 2:3:42 : 3 : 4.

Three friends split $180180 in the ratio 2:3:42:3:4. Total parts =2+3+4=9= 2+3+4 = 9, so one part is $180÷9=$20\$180 \div 9 = \$20.

  • First friend: 2×$20=$402 \times \$20 = \mathbf{\$40}.
  • Second friend: 3×$20=$603 \times \$20 = \mathbf{\$60}.
  • Third friend: 4×$20=$804 \times \$20 = \mathbf{\$80}.

Check: $40+$60+$80=$180\$40 + \$60 + \$80 = \$180. ✓

Concept
Mean Proportional & Continued Proportions

In the proportion ab=bc\frac{a}{b} = \frac{b}{c}, the repeated middle term bb is called the mean proportional (or geometric mean) of aa and cc. Cross-multiplying gives

b2=a⋅c⟹b=a⋅c.b^2 = a \cdot c \qquad\Longrightarrow\qquad b = \sqrt{a \cdot c}.

This is different from the ordinary (arithmetic) average a+c2\frac{a+c}{2}. For example, the mean proportional of 44 and 99 is 4⋅9=36=6\sqrt{4 \cdot 9} = \sqrt{36} = 6, whereas their arithmetic average is 6.56.5.

Concept
The Golden Ratio

The golden ratio appears when a length is split so that the whole is to the larger part as the larger part is to the smaller part. Writing the larger part as aa and the smaller as bb:

a+ba=ab=φ≈1.618.\frac{a+b}{a} = \frac{a}{b} = \varphi \approx 1.618.

Its exact value is φ=1+52\varphi = \dfrac{1+\sqrt{5}}{2}. The golden ratio has a remarkable self-referencing property: φ2=φ+1\varphi^2 = \varphi + 1 and 1φ=φ−1\dfrac{1}{\varphi} = \varphi - 1. It shows up in art, architecture, and the spiral arrangement of leaves and seeds in nature.

Concept
Weighted Averages & Mixtures

A plain average treats every item equally, but a weighted average counts some values more heavily according to a ratio of “weights” (amounts). For two groups,

weighted average=w1v1+w2v2w1+w2,\text{weighted average} = \frac{w_1 v_1 + w_2 v_2}{w_1 + w_2},

where each vv is a value and each ww is how much of it there is. Mixture problems — blending solutions of different concentrations, or averaging test scores of different weights — are exactly this idea. The result always lands between the two values, pulled toward whichever group is larger.

Example
Worked Example: mixing two solutions

Mix 22 liters of a 30%30\% salt solution with 33 liters of an 80%80\% salt solution. What is the concentration of the blend?

concentration=(2)(30%)+(3)(80%)2+3=60+2405=3005=60%.\text{concentration} = \frac{(2)(30\%) + (3)(80\%)}{2 + 3} = \frac{60 + 240}{5} = \frac{300}{5} = \mathbf{60\%}.

Notice 60%60\% sits between 30%30\% and 80%80\%, closer to 80%80\% because there is more of the stronger solution.

Concept
Combined Rates: Work Problems & Average Speed

When two workers (or pipes, or machines) act together, add their rates, not their times. If one finishes a job in aa hours and the other in bb hours, their combined time tt satisfies

1a+1b=1t.\frac{1}{a} + \frac{1}{b} = \frac{1}{t}.

A close cousin appears with average speed over equal distances. If you drive one leg at speed v1v_1 and an equal-length leg at v2v_2, the average speed is not v1+v22\frac{v_1+v_2}{2} but the harmonic mean:

vˉ=2v1v2v1+v2.\bar{v} = \frac{2 v_1 v_2}{v_1 + v_2}.

You spend more time at the slower speed, so the true average is dragged below the plain average.

Example
Worked Example: a work problem and an average-speed problem

(a) Working together. Pipe A fills a tank in 44 hours, pipe B in 66 hours. Together:

14+16=312+212=512 tank per hour  ⟹  t=125=2.4 hours.\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ tank per hour} \;\Longrightarrow\; t = \frac{12}{5} = \mathbf{2.4 \text{ hours}}.

Sensibly, this is faster than either pipe alone.

(b) Average speed. You drive 6060 miles at 3030 mph, then 6060 miles at 6060 mph.

vˉ=2(30)(60)30+60=360090=40 mph.\bar{v} = \frac{2(30)(60)}{30 + 60} = \frac{3600}{90} = \mathbf{40 \text{ mph}}.

The plain average would be 4545 mph, but the correct answer is lower because more time is spent going 3030 mph.

Concept
Scaling Area & Volume

When a figure is enlarged by a scale factor kk, its lengths multiply by kk — but its area multiplies by k2k^2 and its volume by k3k^3:

length×k,area×k2,volume×k3.\text{length} \times k, \qquad \text{area} \times k^2, \qquad \text{volume} \times k^3.

This is why doubling a photo's dimensions uses four times the ink, and why a scale model at 1:101:10 holds only 11000\frac{1}{1000} of the real object's volume. It is the most-missed idea in ratios: side ratios and area (or volume) ratios are not the same number.

Tip

Big-picture tip: Before setting up any advanced ratio problem, ask three questions. (1) Do the quantities keep a constant ratio (direct) or a constant product (inverse)? (2) Am I comparing lengths, areas, or volumes — and do I need kk, k2k^2, or k3k^3? (3) Should I add rates rather than times? Answering these first prevents the most common mistakes.

Try it
Quick check

Fill in the missing number:   5:2=00‾:6\;5:2 = \underline{\hphantom{00}}:6

Work it on paper before revealing the answer.

Show the answer

Answer: 1515

Why It Works

Cross-multiplication is just clearing denominators: multiplying both sides of ab=cd\tfrac ab = \tfrac cd by bdbd gives ad=bcad = bc. Since the step is reversible (divide by bdbd again), ad=bcad = bc holds exactly when the ratios are equal. Proportional reasoning is the same as "y=kxy = kx": the constant kk is the unit rate, and every proportion problem is really finding kk from one pair and applying it to another.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.

Formulas, Proofs & Tips

Tip
The percent relationship
part=percent×whole\text{part} = \text{percent} \times \text{whole}

What it means. "Percent" means "per hundred", so 25%25\% is the number 25100=0.25\tfrac{25}{100}=0.25. Multiplying the whole by that number gives the part.

Example. 20%20\% of 6060 is 0.20×60=120.20\times 60 = 12.

Why it works. p%p\% of WW means pp hundredths of WW, i.e. p100⋅W\tfrac{p}{100}\cdot W. Rearranging the same equation gives the other two questions: percent=partwhole\text{percent}=\tfrac{\text{part}}{\text{whole}} and whole=partpercent\text{whole}=\tfrac{\text{part}}{\text{percent}}.

Tip. Convert the percent to a decimal before multiplying. A percent increase of 20%20\% multiplies by 1.201.20, not 0.200.20 — the 11 keeps the original amount.

Tip
Cross multiplication
ab=cd  ⟺  ad=bc(b,d≠0)\frac{a}{b}=\frac{c}{d} \iff ad = bc \quad (b,d \neq 0)

What it means. Two ratios are equal exactly when their cross products are equal.

Example. 34=x12⇒4x=36⇒x=9\dfrac{3}{4}=\dfrac{x}{12}\Rightarrow 4x=36\Rightarrow x=9.

Why it works. Multiply both sides of ab=cd\tfrac{a}{b}=\tfrac{c}{d} by bdbd: the left becomes ab⋅bd=ad\tfrac{a}{b}\cdot bd = ad and the right becomes cd⋅bd=bc\tfrac{c}{d}\cdot bd = bc.

Tip. Set proportions up so matching units sit in matching positions. If miles are on top on the left, miles go on top on the right.