Demonstration 1 of 4
The order states the question
Is 18 to 12 the same comparison as 12 to 18?
The dashed line marks the denominator package, the measuring stick. The ratio says how many of those sticks the top package fills.
\[18\ \text{oz} / 12\ \text{oz} = 3/2\]
\[12\ \text{oz} / 18\ \text{oz} = 2/3\]
oz means ounces. The first amount goes on top, the second on the bottom. The bottom amount is what counts as one.
Predict first. Switch the order to B compared with A. Will the ratio be bigger or smaller than 1?
Choose an example
Constructed example: the chapter's 18 oz and 12 oz labels, with other Label B amounts.
Calculated values
- Question
- A's amount compared with B's
- Ratio
- 18 oz / 12 oz = 3/2
- Units
- ounces cancel; the ratio has no unit
- Reversed order gives
- 2/3
Divide both by 6: 18 / 6 = 3 and 12 / 6 = 2, so 18 oz / 12 oz = 3/2. A holds more than one B amount: 3/2 of it. The denominator package is the measuring stick, and the lower bar is tiled in copies of it.
Use the idea
When someone says "three to two," ask which comes first. Saying the order out loud tells you which question is being answered.
Where the conclusion applies
Both amounts are in the same unit, so the unit cancels, and the bottom amount is not zero. Comparing ounces with grams would need a conversion first.
Check your understanding: What is the ratio of a 12 oz package to a 6 oz package, and of the 6 oz to the 12 oz?
Chapter 12 source: section "The order states the question". Demonstration C12-D01.
Demonstration 2 of 4
A share of a whole, per hundred
How does 18 out of 24 become a percent?
The top bar is the whole, cut into one cell per response. The bottom bar is the same whole cut into 100. The dashed line shows the share reads the same on both.
\[\frac34=\frac{75}{100}=75\%\]
The part is the number of complete responses, the whole is all responses. Percent means per hundred: 75% is 75/100.
Predict first. Keep 18 complete responses but make the whole 20. Will the percent go up or down?
Choose an example
Constructed example: the chapter's 18 of 24 responses, with other counts and a whole of 20.
Calculated values
- Part
- 18
- Whole
- 24
- Ratio
- 18/24 = 3/4
- Decimal
- 18 / 24 = 0.75
- Percent
- 0.75 x 100 = 75%
- Check
- 0.75 x 24 = 18
18/24: divide both by 6 to get 3/4. As a decimal, 18 / 24 = 0.75, and 0.75 x 100 = 75, so 75%. Check: 0.75 x 24 = 18.
Use the idea
Before turning a count into a percent, say what the whole is. The same 18 is 75% of 24 but 90% of 20.
Where the conclusion applies
Part-to-whole: the part is inside the whole. A part-to-part ratio, like 2 measures of concentrate to 3 of water, needs the total 5 as the whole first.
Check your understanding: Seven of twenty responses are incomplete. What percent is that?
Chapter 12 source: section "Turn a part-to-whole ratio into a percent". Demonstration C12-D02.
Demonstration 3 of 4
Equal percentages, unequal bases
Up 20%, then down 20%: why not back to 100?
Each bar is the quantity after one step. The second change is the same percent as the first, but of a different base, so it is a different amount.
\[100 × 1.20 × 0.80 = 96\]
\[1.20 × 0.80 = 0.96\]
1.20 keeps the whole and adds twenty hundredths. 0.80 keeps eighty hundredths. Each percent is taken of the amount at that step, its base.
Predict first. Change to 50% up, then 50% down. Where does the quantity end?
Choose an example
Constructed example: the chapter's 100 with a 20% rise and fall, with other percents and orders.
Calculated values
- First change (increase)
- 20% of 100 = 20
- Second change (decrease)
- 20% of 120 = 24
- Scale factors
- 1.20 x 0.80 = 0.96
- End
- 100 x 1.20 x 0.80 = 96
- Compared with 100
- 4 below
20% of 100 = 20, so 100 + 20 = 120. The second change uses the new base: 20% of 120 = 24, so 120 - 24 = 96. In one line: 100 x 1.20 x 0.80 = 96, 4 below where it started.
Use the idea
When you read "up 20%" then "down 20%," multiply the scale factors: 1.20 x 0.80 = 0.96, not 1.
Where the conclusion applies
Each percent is taken of the current amount, not the original. If both were taken of the original 100, they would cancel.
Check your understanding: Start at 100, rise 10%, then fall 10%. What is the result?
Chapter 12 source: section "Percent is a scale tied to a base". Demonstration C12-D03.
Demonstration 4 of 4
Price per one ounce
Label A is 18 oz for $4.50 and Label B is 12 oz for $3.24. Which costs less per ounce?
Each bar is a unit price: the label's cents divided by its ounces. Both bars now answer the same question, so the shorter bar is the lower price per ounce.
\[\frac{\$4.50}{18\,\mathrm{oz}}\]
\[450\div18=25\]
\[324\div12=27\]
Prices are written in cents to keep the numbers short: $4.50 is 450 cents, and 25 cents per ounce is $0.25 per ounce. Either is a unit rate: the price of one ounce.
Predict first. Make Label B cost $3.00 for 12 oz. Which label is lower per ounce now?
Choose an example
Constructed example: the chapter's Labels A and B, with other Label B prices and sizes.
Calculated values
- Label A
- $4.50 for 18 oz
- Label B
- $3.24 for 12 oz
- A per ounce
- 450 / 18 = 25 cents
- B per ounce
- 324 / 12 = 27 cents
- Lower unit price
- Label A
Work in cents. A: 450 / 18 = 25 cents per ounce. B: 324 / 12 = 27 cents per ounce. Label A has the lower unit price. Check by multiplying back: 18 x 25 = 450 and 12 x 27 = 324.
Use the idea
To compare two sizes, divide each price by its own amount, then compare the per-one answers. Multiply back to check.
Where the conclusion applies
Constructed labels, not real products. Unit price is the only thing compared; a lower unit price is not a recommendation to buy.
Check your understanding: Fifteen ounces cost $3.60 and ten ounces cost $2.50. Which is lower per ounce?
Chapter 12 source: section "Divide decimal prices with a visible method". Demonstration C12-D04.