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Gears, mixtures and recipes

Ratio, rates and money · level 5, lesson 7 of 7

A question to start

A small gear with 12 teeth meshes with a big gear with 36 teeth. You turn the small gear once, all the way round. How far does the big gear turn?

Now the twist: if you want the big gear to turn faster than the small one, which one should you turn with the motor? And why does a bike have so many gears at all?

Gears, paint, juice and recipes look like four different subjects. They are all ratio, and this lesson puts together everything from this level: proportions, direct and inverse proportion, and percent.

The idea

Gears: teeth go past at the same rate

A gear is a wheel with teeth round its edge. When two gears mesh (their teeth fit together), turning one turns the other, in the opposite direction.

12 teeth 36 teeth

Here is the key idea. Where the gears touch, each tooth of one gear pushes exactly one tooth of the other. So the same number of teeth go past the meeting point on both gears.

  • One full turn of the small gear moves 12 of its teeth past the meeting point.
  • So 12 teeth of the big gear go past too. The big gear has 36 teeth, so it turns \(\dfrac{12}{36} = \dfrac13\) of a turn.

For two meshing gears,

\[\text{turns}_1 \times \text{teeth}_1 = \text{turns}_2 \times \text{teeth}_2\]

because both sides count the teeth that went past. The product stays the same, so turns are inversely proportional to teeth (lesson 2): a gear with three times as many teeth turns one third as many times.

The gear ratio compares the teeth: here 12 : 36, or 1 : 3. The small gear turns 3 times for every 1 turn of the big gear.

Speed or strength. If the motor turns the small gear, the big gear turns slowly but with more turning force (this is how a cyclist climbs a hill, or how a winch lifts a heavy load). If the motor turns the big gear, the small gear spins three times as fast but with less force. You trade one for the other; gears never give both for free.

Bikes

On a bike, a chain joins the gear at the pedals (the chainring) to a gear on the back wheel (the cog). The chain does the job of the meeting point: every link that leaves one gear arrives at the other, so the same teeth-count rule holds. With a 48-tooth chainring and a 16-tooth cog, one turn of the pedals pulls 48 links, which turns the back wheel \(48 \div 16 = 3\) times.

  • A big chainring and a small cog: many wheel turns per pedal turn. Fast on flat roads, hard to push.
  • A small chainring and a big cog: few wheel turns per pedal turn. Slow, but easy for climbing hills.

Gear trains

With three gears in a row, A meshes with B and B meshes with C. The teeth that pass from A to B are the same teeth that pass from B to C, so

\[\text{turns}_A \times \text{teeth}_A = \text{turns}_C \times \text{teeth}_C.\]

The middle gear (called an idler) does not change the ratio between A and C. It only changes the direction: A and C turn the same way.

Mixtures: part to part, part to whole

A mixture combines ingredients in a ratio. Paint mixed 2 parts blue to 3 parts white means for every 2 cups (or litres, or spoonfuls) of blue there are 3 of the same unit of white.

  • Part to part: blue : white \(= 2 : 3\).
  • Part to whole: there are \(2 + 3 = 5\) parts in all, so blue is \(\dfrac25\) of the paint and white is \(\dfrac35\).

To make a certain amount, find the size of one part first (sharing in a ratio). For 15 litres: 5 parts make 15 litres, so 1 part is 3 litres; blue is \(2 \times 3 = 6\) litres and white is \(3 \times 3 = 9\) litres.

Why the colour stays the same when you scale. The shade depends only on the fraction of blue, \(\dfrac25\). Doubling both parts to 4 : 6 keeps that fraction (\(\dfrac{4}{10} = \dfrac25\)), so it is the same colour. Adding 1 to both parts to 3 : 4 does not: \(\dfrac37 \ne \dfrac25\), a different colour.

Concentration in percent

The concentration of an ingredient is its part-to-whole ratio written as a percent. Juice made from 1 part concentrate to 4 parts water is

\[\frac{1}{1 + 4} = \frac15 = 20\% \text{ concentrate.}\]

Careful: the ratio is 1 : 4, but the concentration is 1 out of 5, not 1 out of 4. A part-to-part ratio leaves out the part on top when it counts the total.

Mixing two mixtures

Mix 200 mL of a 30% juice with 300 mL of a 10% juice. Is the result 20% (the average of 30 and 10)? Count the actual juice instead:

  • From the first: \(0.30 \times 200 = 60\) mL of juice.
  • From the second: \(0.10 \times 300 = 30\) mL of juice.
  • Total: 90 mL of juice in 500 mL of drink, which is \(\dfrac{90}{500} = 18\%\).

It is less than 20% because there is more of the weak juice. The rule: add up the amounts of the ingredient, then divide by the total. Never average the percents unless the two amounts are equal. (This is a weighted average, which you will meet again in Statistics.)

Recipes

A recipe is a mixture with many ingredients. To scale it, multiply every ingredient by the same scale factor: \(\dfrac{\text{servings wanted}}{\text{servings in recipe}}\). Things that are not ingredients do not scale that way: the oven temperature stays the same, and the baking time changes only a little (a bigger cake takes longer, but not twice as long).

Worked examples

1. Two gears. A 12-tooth gear drives a 36-tooth gear. The small gear turns 60 times a minute. How fast does the big gear turn?

  • Teeth per minute: \(60 \times 12 = 720\). The big gear turns \(720 \div 36 =\) 20 times a minute.
  • Check with the ratio: the big gear has 3 times the teeth, so it turns \(\tfrac13\) as often. ✓

2. A bike. A bike has a 48-tooth chainring and a 16-tooth cog. The back wheel goes 2.1 m along the ground for each turn. How far does the bike go for one turn of the pedals?

  • Wheel turns per pedal turn: \(48 \div 16 = 3\).
  • Distance: \(3 \times 2.1 =\) 6.3 m.

3. A gear train. Gear A (10 teeth) drives gear B (30 teeth), which drives gear C (15 teeth). A turns 6 times. How many times does C turn?

  • Teeth passed: \(6 \times 10 = 60\). These 60 teeth go through B to C.
  • C turns \(60 \div 15 =\) 4 times. (B's 30 teeth do not matter; B turns 2 times.)

4. Paint. Paint is mixed 2 parts blue to 3 parts white. How much of each for 15 litres? What percent is blue?

  • 5 parts make 15 L, so 1 part is 3 L. Blue \(= 6\) L, white \(= 9\) L.
  • Blue is \(\dfrac{6}{15} = \dfrac25 =\) 40%.

5. Mixing juices. 200 mL of 30% juice is mixed with 300 mL of 10% juice. Concentration?

  • Juice: \(60 + 30 = 90\) mL in 500 mL, so \(\dfrac{90}{500} =\) 18%.

6. Scaling a recipe. A pasta recipe for 4 people uses 300 g of pasta and \(\tfrac34\) cup of cream. Scale it for 6.

  • Scale factor \(\dfrac{6}{4} = 1.5\).
  • Pasta: \(300 \times 1.5 = 450\) g. Cream: \(\tfrac34 \times 1.5 = \tfrac98 = 1\tfrac18\) cups.

Try it

  1. A 15-tooth gear meshes with a 45-tooth gear. The big gear turns 10 times. How many times does the small gear turn?
  2. A 40-tooth gear drives a 10-tooth gear. The big gear turns 25 times a minute. How fast does the small gear turn?
  3. A bike has two set-ups: 52 teeth at the front with 13 at the back, or 39 at the front with 26 at the back. Find the wheel turns per pedal turn for each. Which is better for a steep hill?
  4. Paint is mixed 3 parts red to 5 parts yellow. How much of each for 24 litres? What percent is red?
  5. A drink is 1 part concentrate to 5 parts water. What percent is concentrate? How much concentrate is in 1.2 litres of the drink?
  6. 300 mL of 20% juice is mixed with 200 mL of 50% juice. What is the concentration of the mixture?
  7. A recipe for 12 cookies uses \(\tfrac34\) cup of sugar. How much sugar for 30 cookies?
  8. You have 400 mL of 30% juice. How much water must you add to make it 20% juice?
Answers
  1. Teeth passed: \(10 \times 45 = 450\). Small gear: \(450 \div 15 = 30\) turns.
  2. Teeth per minute: \(25 \times 40 = 1000\). Small gear: \(1000 \div 10 = 100\) turns a minute.
  3. \(52 \div 13 = 4\) wheel turns per pedal turn; \(39 \div 26 = 1.5\). The 39 and 26 set-up is better for a hill: each push of the pedals turns the wheel less, so it needs less force.
  4. 8 parts make 24 L, so 1 part is 3 L. Red \(= 9\) L, yellow \(= 15\) L. Red is \(\dfrac38 = 37.5\%\).
  5. \(\dfrac{1}{6} \approx 16.7\%\) concentrate. In 1.2 L: 6 parts make 1.2 L, so 1 part (the concentrate) is \(1.2 \div 6 = 0.2\) L, which is 200 mL.
  6. Juice: \(0.20 \times 300 = 60\) mL and \(0.50 \times 200 = 100\) mL. Total 160 mL in 500 mL: \(\dfrac{160}{500} = 32\%\).
  7. Scale factor \(\dfrac{30}{12} = 2.5\). Sugar: \(\tfrac34 \times 2.5 = \tfrac{15}{8} = 1\tfrac78\) cups.
  8. The juice does not change: \(0.30 \times 400 = 120\) mL. For 120 mL to be 20% of the drink, the drink must be \(120 \div 0.20 = 600\) mL. So add \(600 - 400 = 200\) mL of water.

Common mistake

Thinking the bigger gear turns more. The gear with more teeth turns fewer times: turns and teeth are inversely proportional. Count the teeth that go past, and the direction is clear.

Averaging percents. Mixing 30% juice with 10% juice does not give 20% unless the amounts are equal. Add the amounts of juice, then divide by the total.

Using part-to-part as part-to-whole. A 1 : 4 mix is 1 part in 5, which is 20%, not 25%.

Where it leads