Unit price and comparing rates

When you reach for it

Tova needs batteries. A pack of 8 costs 6 dollars and a pack of 12 costs 8.40 dollars. The bigger pack costs more, but it also holds more, so neither price settles the question on its own. What settles it is the price of one battery in each pack.

The price of one item, or of one unit of measure such as one liter, is the unit price. It is the Rates and unit rates of a price: dollars per battery, dollars per liter. Reach for it whenever the same thing is sold in different sizes and you want to know which size gives more for the money.

The same move compares any two rates, not only prices: which of two pumps fills a tank faster, which of two printers is quicker. Bring each rate down to one unit, then compare the two single numbers. Part 1 (day one) finds and compares unit prices; Part 2 (day two) compares through the other unit rate, how much per dollar, and compares rates that are not prices at all.

The procedure

Part 1: the price of one

A unit price is a price shared out equally over the items, so it is found by division, the way every unit rate is:

For Tova’s pack of 8, that is 6 \div 8 = 0.75 dollars per battery, or 75 cents. Sharing 6 dollars equally over 8 batteries is exactly what 6 \div 8 means, which is why division is the right move. The units travel with the answer: dollars per battery.

The “unit” can be any fixed amount the shopper picks. Detergent is sold by the liter, so its unit price is dollars per liter: a 3 liter bottle at 10.50 dollars costs 10.50 \div 3 = 3.50 dollars per liter. Shampoo comes in milliliters, and the price of a single milliliter is a tiny number, so it is easier to use 100 milliliters as the unit. A 250 milliliter bottle is 2.5 lots of 100 milliliters, so at 4 dollars it costs 4 \div 2.5 = 1.60 dollars per 100 milliliters. Dividing by a decimal is ordinary decimal division.

Part 1: comparing two sizes

Find the unit price of each size in the same unit, then compare the two numbers. The lower unit price is the better buy: every battery, or every liter, costs less, so any amount bought at that rate costs less.

For Tova’s packs, the pack of 12 costs 8.40 \div 12 = 0.70 dollars per battery against 0.75 dollars in the pack of 8. Figure 1 shows both packs on double number lines drawn to the same scale. Over the mark for 1 battery, the pack of 12 reads 0.70 and the pack of 8 reads 0.75.

This is the method of Deciding which of two ratios is larger with a particular choice of common amount: both ratios of dollars to batteries are scaled to 1 battery. Any common amount gives the same verdict. Scaling both to 24 batteries gives 3 \times 6 = 18 dollars against 2 \times 8.40 = 16.80 dollars, and the pack of 12 still wins. The unit price is the common amount that works for every pair at once, which is why a shop shelf can print one on every label.

Part 1: rounding to the cent

A unit price often does not come out to a whole number of cents. A pack of 3 batteries for 2 dollars costs 2 \div 3 = 0.666\ldots dollars each, which rounds to 0.67 dollars. Rounding to the nearest cent is usually enough to decide, and there is a clean test for when it is.

If the two rounded prices are different, the order is certain. Anything that rounds to 0.74 is below 0.745, and anything that rounds to 0.75 is at least 0.745, so the first is the lower price. (The true gap may be much smaller than the one cent the rounded prices suggest.)

If the two rounded prices are equal, rounding has hidden the difference. Carry the division one more place, to tenths of a cent, or compare the cost of an amount both sizes can make, as above. Example 3 is such a case.

Part 2: how much per dollar

Every rate gives two unit rates, one the flip of the other (Rates and unit rates). For a price, the other one is how much per dollar: milliliters per dollar, batteries per dollar. It is found by dividing the other way, amount divided by price. A 400 milliliter bottle at 5 dollars gives 400 \div 5 = 80 milliliters per dollar.

Read this one the other way round: more per dollar is the better buy, since each dollar brings home more. The two unit rates always agree on which buy is better, because they are reciprocals. If each dollar buys more milliliters, then each milliliter takes a smaller share of a dollar. Choose whichever division is easier. Often that is the one that divides by the price, when the price is a round number of dollars.

Part 2: rates that are not prices

Nothing above needed money. To compare two pumps, find the liters each moves in one minute: the pump with more liters per minute is faster. To compare two printers, find the pages each prints in one minute: more pages per minute is quicker. The other unit rate works too, with the reading turned round: fewer minutes per liter, or fewer seconds per page, is faster.

Two rates can only be compared when they are written the same way, with the same units. If one printer is described in pages per minute and the other in seconds per page, turn one of them round first, and change minutes to seconds (or back) so the units match.

Worked examples

Examples 1 to 4 go with Part 1, Examples 5 to 7 with Part 2.

Boundaries

Same thing, same unit. A unit price compares sizes of the same product. Both prices must be per the same unit, both per liter or both per 100 milliliters, never one of each.

Nothing is per zero. A unit price divides by the amount, so the amount cannot be 0.

The cheaper rate is not always the better choice. A lower unit price saves money only on an amount you will actually use, so the big size is no bargain if most of it goes to waste.

Equal cents do not mean equal prices. When two unit prices round to the same cent, carry one more place or compare a common amount before calling them equal.

Higher or lower depends on the order. In dollars per item, lower is better; in items per dollar, higher is better. In liters per minute, higher is faster; in minutes per liter, lower is faster. The units say which reading applies.

Where this goes next

Comparing through one unit is the everyday form of comparing ratios. A shop that prices one bottle in dollars per liter and another in dollars per gallon needs the units of a rate changed before the two can be compared. When the amounts are fractions, as in \displaystyle \frac{3}{4} liter for 2.40 dollars, the unit price comes from dividing by a fraction. The pump rates here lead to flow rate, and a price of so much per item becomes the The constant of proportionality and proportional tables, the number that turns any amount into its cost.

Exercises

Part 1: finding and comparing unit prices

1. A pack of 4 batteries costs 3.60 dollars. Find the unit price.
2. A 4 liter bottle of detergent costs 13.60 dollars. Find the price per liter.
3. A 350 milliliter bottle of shampoo costs 4.90 dollars. Find the price per 100 milliliters.
4. Which is the better buy: 6 batteries for 4.80 dollars or 10 batteries for 7.50 dollars?
5. Which is the better buy: 2 liters of detergent for 6.80 dollars or 5 liters for 16 dollars?
6. Which is the better buy: 200 milliliters of shampoo for 3.20 dollars or 500 milliliters for 7.50 dollars?
7. A pack of 6 batteries costs 4.99 dollars. Find the unit price to the nearest cent.
8. Find each unit price to the nearest cent and say which is the better buy: 12 batteries for 9.99 dollars or 20 batteries for 15.99 dollars.
9. Find each unit price to the nearest cent: 6 batteries for 5.29 dollars and 9 batteries for 7.95 dollars. Which is the better buy?
10. A 5 liter bottle of detergent costs 16 dollars. What would a 2 liter bottle have to cost to have the same unit price?
11. Batteries come in packs of 4 for 3 dollars, 10 for 7.20 dollars and 24 for 18 dollars. Find each unit price. Which pack is the best buy?

Part 2: the other unit rate, and rates that are not prices

12. A 500 milliliter bottle of shampoo costs 4 dollars. How many milliliters does each dollar buy?
13. A pack of 18 batteries costs 9 dollars. Find the batteries per dollar and the dollars per battery.
14. Compare by milliliters per dollar: 350 milliliters of shampoo for 5 dollars or 600 milliliters for 8 dollars. Which is the better buy?
15. Pump A moves 150 liters in 6 minutes and pump B moves 168 liters in 7 minutes. Which is faster?
16. Pump A moves 32 liters per minute and pump B moves 90 liters in 3 minutes. How long does each take to fill a 600 liter tank?
17. Printer P prints 60 pages in 4 minutes and printer Q prints 39 pages in 3 minutes. Which is quicker, and how long does each take to print 195 pages?
18. Printer R prints 20 pages per minute. Printer S takes 2.5 seconds per page. Which is quicker? Give both rates in pages per minute and in seconds per page.
19. Printer T takes \displaystyle \frac{1}{4} minute per page. Printer V prints 5 pages in \displaystyle 1\frac{1}{2} minutes. Which is quicker?
20. One pump moves 45 liters in \displaystyle \frac{3}{4} minute and another moves 50 liters in \displaystyle \frac{5}{6} minute. Which is faster?

Reasoning

21. Shampoo X gives 90 milliliters per dollar and shampoo Y gives 75 milliliters per dollar. Without dividing again, which has the lower price per milliliter? Explain.
22. Two unit prices round to 0.42 dollars and 0.43 dollars. Explain why the first is certainly the lower price, and why two unit prices that both round to 0.42 dollars could be in either order.
23. Is the largest pack always the best buy? Use the packs in Exercise 11 to explain.
24. For 6 batteries at 4.80 dollars and 10 at 7.50 dollars, compare the cost of 30 batteries from each. Explain why comparing unit prices must give the same verdict.