Unit Prices and the Metric System
At the deli counter, the sign says cherries cost $\$5.80$ per kilogram. You scoop up a bag that the scale reads as $750$ grams. How much do you pay? The price is for kilograms and the bag is in grams, so you have to deal with the mismatch first.
Price per Unit
Why multiply? Two meters of ribbon cost twice as much as one meter, and 2.4 meters cost 2.4 times as much. Every extra unit adds the same price.
Try the next widget. Slide the unit price and watch how the total grows with the amount.
At the local grocery store, the price of apples is $1,80 \text{ \$}$ per kilogram. If we want to buy a bag that weighs exactly $500 \text{ g}…
At the local grocery store, the price of apples is $1,80 \text{ \$}$ per kilogram. If we want to buy a bag that weighs exactly $500 \text{ g}$, how much money will we have to pay?
A $1 \text{ L}$ bottle of olive oil costs $8,40 \text{ \$}$. If a restaurant buys a $5 \text{ L}…
A $1 \text{ L}$ bottle of olive oil costs $8,40 \text{ \$}$. If a restaurant buys a $5 \text{ L}$ bottle that maintains the same price per liter, how much will they pay for the whole bottle?
Convert Before You Multiply
Almost every mistake in these problems comes from multiplying before the quantity is in the same unit as the price. To convert, use the metric "staircase."
Metric units are built from a base unit (meter, gram, liter) and a prefix. Each prefix is $10$ times the next one down.
The same prefixes work for mass (kg, hg, dag, g, dg, cg, mg) and for capacity (kL, hL, daL, L, dL, cL, mL). A metric ton (t) is $1{,}000$ kg.
In a stationery store, a gram of special craft paper costs $0,02 \text{ \$}$. If we buy a sheet that weighs $80 \text{ dg}$, how much will we have to pay?
In a stationery store, a gram of special craft paper costs $0,02 \text{ \$}$. If we buy a sheet that weighs $80 \text{ dg}$, how much will we have to pay?
At your local grocery store, a kilogram of cherries costs $\$5,80$. If you decide to buy a bag containing $750 \text{ g}$ of cherries and hand over a $\$10$ bill to pay,…
At your local grocery store, a kilogram of cherries costs $\$5,80$. If you decide to buy a bag containing $750 \text{ g}$ of cherries and hand over a $\$10$ bill to pay, how much change will you get back?
A delivery van can transport a maximum of $1,5 \text{ t}$ (metric tons). If the transport cost is $0,10 \text{ \$}…
A delivery van can transport a maximum of $1,5 \text{ t}$ (metric tons). If the transport cost is $0,10 \text{ \$}$ per kilogram of cargo, how much will the delivery cost if the van is full to its maximum capacity?
Prices with Length
When the price is per meter, convert every length to meters and multiply. If there are several pieces, convert them all first and then add.
A fence goes around a rectangle, so its length is the perimeter: $P = 2\ell + 2w$. If part of the side is a gate that you already own, subtract that gap before you multiply by the price.
For an art project, Ethan needs to buy $2 \text{ m}$ and $40 \text{ cm}$ of decorative ribbon. If a meter of ribbon costs $0,50 \text{ \$}…
For an art project, Ethan needs to buy $2 \text{ m}$ and $40 \text{ cm}$ of decorative ribbon. If a meter of ribbon costs $0,50 \text{ \$}$, what is the total cost of the ribbon Ethan needs?
A farmer wants to fence a rectangular plot for his sheep that measures $1,5 \text{ hm}$ long and $80 \text{ m}$ wide. A meter of wire fencing costs $\$6,40…
A farmer wants to fence a rectangular plot for his sheep that measures $1,5 \text{ hm}$ long and $80 \text{ m}$ wide. A meter of wire fencing costs $\$6,40$. In addition, he must leave a gap of $30 \text{ dm}$ to install a gate he has already bought. What will be the total budget to buy the necessary fencing?
In a hardware store, network cable is sold at $\$0,85$ per meter. You need to buy several pieces to install internet at home: one of $1250 \text{ cm}$, another of $85…
In a hardware store, network cable is sold at $\$0,85$ per meter. You need to buy several pieces to install internet at home: one of $1250 \text{ cm}$, another of $85 \text{ dm}$ and a final piece of $4 \text{ m}$. If the store clerk gives you a discount of $\$2,50$ for buying more than $20 \text{ m}$ in total, how much will you have to pay in the end?
Prices with Area
When the price is per square meter, find the area in $\text{m}^2$ and multiply. Convert the side lengths to meters first. Then meters times meters give square meters directly.
Converting areas directly is different. A square that is $1$ m on each side is $10$ dm on each side, so its area is $10 \times 10 = 100$ square decimeters. Each step on the staircase is $\times 100$, not $\times 10$.
| Unit | Equals |
|---|---|
| $1 \text{ m}^2$ | $100 \text{ dm}^2 = 10{,}000 \text{ cm}^2$ |
| $1 \text{ dam}^2$ | $100 \text{ m}^2$ |
| $1 \text{ hm}^2 = 1$ hectare (ha) | $100 \text{ dam}^2 = 10{,}000 \text{ m}^2$ |
| $1 \text{ km}^2$ | $100 \text{ hm}^2 = 1{,}000{,}000 \text{ m}^2$ |
Sophia's mother has bought a rectangular rug for her bedroom that measures $3 \text{ m}$ long and $2 \text{ m}$ wide. If the price of the rug is $15 \text{ \$}…
Sophia's mother has bought a rectangular rug for her bedroom that measures $3 \text{ m}$ long and $2 \text{ m}$ wide. If the price of the rug is $15 \text{ \$}$ per square meter ($m^2$), how much did the rug cost?
You want to renovate the floor of your bedroom, which has a rectangular shape with dimensions of $4 \text{ m}$ long and $350 \text{ cm}$ wide. The price of the laminate…
You want to renovate the floor of your bedroom, which has a rectangular shape with dimensions of $4 \text{ m}$ long and $350 \text{ cm}$ wide. The price of the laminate flooring you like is $\$18,50$ per square meter. What will be the total cost of the material to cover the whole floor?
A farmer has inherited a plot of rural land of $2,4 \text{ ha}$ (hectares). Knowing that the price of land in that area is $\$12$ per square decameter ($\text{dam}^2…
A farmer has inherited a plot of rural land of $2,4 \text{ ha}$ (hectares). Knowing that the price of land in that area is $\$12$ per square decameter ($\text{dam}^2$), what is the total value of the farm?
Prices with Capacity and Volume
For liquids, the price is usually per liter. These facts connect capacity and volume:
-
$1 \text{ L} = 1{,}000 \text{ mL}$
-
$1 \text{ dm}^3 = 1 \text{ L}$
-
$1 \text{ cm}^3 = 1 \text{ mL}$
-
$1 \text{ m}^3 = 1{,}000 \text{ L}$
Volume units go in steps of $1{,}000$, because $10 \times 10 \times 10 = 1{,}000$.
A farmer sells honey in $250 \text{ ml}$ jars at a price of $4 \text{ \$}$ each. If a customer wants to buy exactly $1 \text{ L}…
A farmer sells honey in $250 \text{ ml}$ jars at a price of $4 \text{ \$}$ each. If a customer wants to buy exactly $1 \text{ L}$ of honey, how many jars do they need and how much money will they spend in total?
For a birthday party, you need to buy $12 \text{ L}$ of orange drink. In the supermarket you have two options: $1,5 \text{ L}$ bottles at $\$1,20…
For a birthday party, you need to buy $12 \text{ L}$ of orange drink. In the supermarket you have two options:
-
$1,5 \text{ L}$ bottles at $\$1,20$ each.
-
6-can packs of $33 \text{ cl}$ at $\$3,15$ per pack.
Calculate how much it would cost you to buy exactly $12 \text{ L}$ using only the first option. How many packs of cans would you have to buy to have at least that amount?
A tanker truck transports $15 \text{ m}^3$ of drinking water to supply a drought-stricken area. a) If the water is sold at $\$0,05$ per decaliter ($\text{daL}…
A tanker truck transports $15 \text{ m}^3$ of drinking water to supply a drought-stricken area.
- a) If the water is sold at $\$0,05$ per decaliter ($\text{daL}$), how much is the truck's entire load worth?
- b) How many $5 \text{ L}$ bottles could be filled with all that volume?
Comparing Deals
Which is the better buy, a small package or a big one? To compare fairly, find the price of the same amount of each, usually one unit.
If the quantity is a simple fraction of a unit, you can also scale up. A $250$ g bag is a quarter of a kilogram, so $4$ bags make $1$ kg and the price per kilogram is $4$ times the price of one bag.
A label like "$20\%$ extra free" means the package holds $120\%$ of the usual mass for the usual price.
A $250 \text{ g}$ bag of coffee costs $3 \text{ \$}$. What is the price per kilogram of this same coffee?
A $250 \text{ g}$ bag of coffee costs $3 \text{ \$}$. What is the price per kilogram of this same coffee?
A perfume shop is offering a promotion on your favorite cologne. The $75 \text{ ml}$ bottle costs $\$13,50$, while the "economy" size bottle of $0,2 \text{ L}…
A perfume shop is offering a promotion on your favorite cologne. The $75 \text{ ml}$ bottle costs $\$13,50$, while the "economy" size bottle of $0,2 \text{ L}$ costs $\$32$.
- a) What is the price per liter in each of the formats?
- b) Which format is cheaper?
In a supermarket we find two offers for jars of jam of the same brand: Option A: A $450 \, g$ jar that costs $\$2,85 \, \$$. Option B: A $0,6 \, kg$ jar with an offer of…
In a supermarket we find two offers for jars of jam of the same brand:
-
Option A: A $450 \, g$ jar that costs $\$2,85 \, \$$.
-
Option B: A $0,6 \, kg$ jar with an offer of "$20 \, \%$ extra free" (i.e. the jar contains $20 \, \%$ more mass but the price is that of the standard $600 \, g$ jar), for a price of $\$3,90 \, \$$.
Determine which of the two options is cheaper by calculating the price per kilogram of each.
Mass, Volume, and Density
Some problems give you a volume of liquid but charge you by mass. Density connects the two.
Since $1 \text{ mL} = 1 \text{ cm}^3$ and $1 \text{ L} = 1 \text{ dm}^3$, these densities are the same number: $1 \text{ g/cm}^3 = 1 \text{ kg/dm}^3 = 1 \text{ kg/L}$. Water has density $1$.
Setting a selling price
A business doesn't sell a product for what the ingredients cost. The selling price has to cover every cost of making one item, plus the profit the business wants to keep.
Costs come in different forms. The liquid is priced by mass, while the bottle and the box are priced per piece. Find the cost of each part for one item, then add them.
Other problems give a weight limit and ask how many items fit. Since you can't load part of a box, round down.
A cosmetics company manufactures a luxury perfume. To fill each $75 \, ml$ bottle, they use an essence with a density of $0,85 \, g/cm^3$. The production cost of the…
A cosmetics company manufactures a luxury perfume. To fill each $75 \, ml$ bottle, they use an essence with a density of $0,85 \, g/cm^3$. The production cost of the essence is $\$145 \, \$$ per kilogram. In addition, the glass bottle costs $\$2,15 \, \$$ and the packaging $\$0,80 \, \$$. If they want to make a profit of $\$15 \, \$$ on each bottle sold, what should the retail price of a single bottle be?
A delivery truck has a maximum load capacity of $3,5 \, t$. It must transport boxes of concentrated detergent. Each box contains $24$ bottles, and each bottle has a mass…
A delivery truck has a maximum load capacity of $3,5 \, t$. It must transport boxes of concentrated detergent. Each box contains $24$ bottles, and each bottle has a mass of $1.250 \, g$. If each empty box weighs $800 \, g$ and the selling price of each individual bottle is $\$6,45 \, \$$, what is the total value of the goods transported by the truck when it is loaded to its maximum capacity without exceeding it?
A cube-shaped tank with an edge of $1,5 \, m$ is full of olive oil. We know that the density of the oil is $0,92 \, kg/dm^3$. If the oil is bottled in $75 \, cl$ bottles…
A cube-shaped tank with an edge of $1,5 \, m$ is full of olive oil. We know that the density of the oil is $0,92 \, kg/dm^3$. If the oil is bottled in $75 \, cl$ bottles and each bottle is sold on the international market at a price of $\$12,50 \, $, what is the total value of the tank in dollars? (Assume the tank is completely emptied).
Multi-Step Problems with Surface Area and Cost
Painting a room takes several steps. Find the area to be painted, change area to liters of paint, change liters to cans, and finally multiply by the price. Cans are sold whole, so round up.
The four walls of a room, rolled out flat, form one long rectangle. Its length is the perimeter of the floor and its height is the height of the room. A ceiling is just an extra $\ell \times w$.
Sometimes you compare two price formulas, one with a flat fee. Put both prices in the same unit: $1 \text{ m}^2 = 10{,}000 \text{ cm}^2$, so a price per $\text{cm}^2$ becomes a price per $\text{m}^2$ when you multiply by $10{,}000$.
We want to paint the four walls and the ceiling of an exhibition room that has the shape of a cuboid. The dimensions of the base are $12 \, m$ long by $80 \, dm$ wide,…
We want to paint the four walls and the ceiling of an exhibition room that has the shape of a cuboid. The dimensions of the base are $12 \, m$ long by $80 \, dm$ wide, and the height of the room is $350 \, cm$. We know that:
-
One of the long walls has a glass panel that occupies $15 \, m^2$.
-
The coverage of the paint is $1 \, L$ for every $10 \, m^2$.
-
The paint is sold in $5 \, L$ cans at a price of $\$42,50 \, \$$ per can.
Calculate the total cost of the paint needed, bearing in mind that only full cans can be bought.
Artificial grass is to be installed on a training field measuring $0,4 \, hm$ long and $250 \, dm$ wide. There are two quotes: Company "TotalGreen": $\$18,50 \, \$$ per…
Artificial grass is to be installed on a training field measuring $0,4 \, hm$ long and $250 \, dm$ wide. There are two quotes:
-
Company "TotalGreen": $\$18,50 \, \$$ per square meter, but they charge a fixed installation fee of $\$450 \, \$$.
-
Company "MeadowPlus": $\$0,0022 \, \$$ per square centimeter, with installation included in the price.
Which company is cheaper for this field and how much money is saved compared to the other option?
Multi-Step Problems with Volume of Water
Water bills are charged per $\text{m}^3$. If a water level drops, the water that is lost is a rectangular prism whose height is the drop in the level. Volume is base area times height, and the units must all be meters before you multiply.
Some problems give a rate per square meter. Then multiply by the area and convert liters to $\text{m}^3$ at the end.
An organic farmer has a citrus orchard that occupies an area of $0,85 \, ha$. To keep the trees hydrated during the summer, they use an irrigation system that consumes…
An organic farmer has a citrus orchard that occupies an area of $0,85 \, ha$. To keep the trees hydrated during the summer, they use an irrigation system that consumes exactly $12 \, L$ of water per square meter per day. If the price of irrigation water is $\$0,35 \, \$$ per cubic meter ($m^3$), what will the water bill amount to after a month of $30$ days?
A municipal swimming pool $25 \, m$ long and $12,5 \, m$ wide has been filled to a height of $180 \, cm$. Due to evaporation and use, the water level drops by $5 \, mm$…
A municipal swimming pool $25 \, m$ long and $12,5 \, m$ wide has been filled to a height of $180 \, cm$. Due to evaporation and use, the water level drops by $5 \, mm$ every day. If the price of water is $\$1,95 \, \$$ per cubic meter ($m^3$), calculate how much money it costs the city council to replace the lost water after a summer season of $90$ days.
Back to the start: $750$ g is $0.75$ kg, and $5.80 \times 0.75 = \$4.35$ for the cherries.