Problem
Percent change in a price
A pack of noodles went from 0.80 to 1.00 dollars. A carton of milk went from 2.00 to 2.40 dollars. Decide which one rose more.
Reasoning · Grade 6-1 Ratio and Rate (1)
Your answer
Watch out The bigger money rise looks like more, but it must be weighed against the old price.
How to solve
Strategy Analyze the Units — Subtracting gives 20 cents for the noodles and 40 cents for the milk, and it is tempting to stop there. Looking at the units stops that: cents alone is not a fair measure of a price rise, because the two items are not the same size purchase to begin with. The honest unit is cents of rise per dollar of old price — a rate, which written out of 100 is the percent increase. I draw each old price as a bar with the rise stacked on top so the two rises can be seen as fractions of their own bars rather than as raw lengths. Then I check the conclusion with an easier version of the same question: rescale the milk so it starts at $0.80 too, and see which rise is bigger when both start from the same place.
1STEP 1
Find how many cents each item rose
The rises are 20 and 40 cents.
0.80 = 2.40 - 0.40
Subtracting two prices to get the amount of the change is the first thing to do, but it answers 'how many cents', not yet 'how much of a rise'.
5.NBT.B.7Analyze The Units2STEP 2
Ask what those cents should be compared to
Compare each rise against its old price.
(percent increase) = (rise)/(old price) × 100
Whenever two changes start from different sized amounts, dividing each change by its own starting amount puts them into the same unit so they can be compared at all.
6.RP.A.3Analyze The UnitsEach rise has to be compared against its own starting price, not against the other rise.
Why?
A rise means nothing on its own until it is set against the amount it grew from, so the starting price is the whole it is a part of.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
Why?
Dividing each rise by its own starting price puts both into the same unit, and only then can they be placed in order.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
3STEP 3
Draw the two rises as parts of their own bars
As bars they are 1/4 and 1/5.
0.20/0.80 = 1/4, 0.40/2.00 = 1/5
The picture makes the trap visible: the longer add-on belongs to the longer bar, so length by itself cannot decide which price rose more steeply.
6.RP.A.3Draw A Diagram4STEP 4
Turn each fraction into a percent
As percents they are 25 % and 20 %.
0.20/0.80 × 100 = 25 %, 0.40/2.00 × 100 = 20 %
Both fractions simplify to easy unit fractions, so the percents can be read off without long division: a quarter is 25 out of 100 and a fifth is 20 out of 100.
7.RP.A.3Analyze The Units5STEP 5
Compare the two rates
So the noodles rose more.
25 % ＞ 20 %
Once both changes are expressed as percents of their own starting prices, they are two numbers in the same unit and can simply be put in order.
7.RP.A.3Analyze The Units6STEP 6
Confirm it by putting both on the same starting price
Putting both on one starting price gives the same answer.
0.16 ＜ $0.20
Rescaling both items to a common starting price removes the only thing that made the comparison tricky, and the winner does not change when you rescale.
7.RP.A.3Solve An Easier Related ProblemAnswer
the noodles
25 %, 20 %
Both answers are percents between 0 and 100, which is what a moderate price rise should look like, and neither is anywhere near a doubling. The two figures can be checked straight from the prices: 25 % of 0.80 is 0.20, and 0.80 + 0.20 = 1.00, the printed new noodle price; 20 % of 2.00 is 0.40, and 2.00 + 0.40 = 2.40, the printed new milk price. The answer is a choice between two named items, as the question asks, and it is deliberately not the item with the bigger cash rise.
Takeaway
Bigger in cents is not the same as bigger as a price rise — always divide the rise by the price it started from.
- Find how many cents each item rose
- Ask what those cents should be compared to
- Draw the two rises as parts of their own bars
- Turn each fraction into a percent
- Compare the two rates
- Confirm it by putting both on the same starting price