Price Problems
At the street market, a kilogram of tomatoes costs $\$28$ and you have a bag of $750\text{ g}$. At the hardware store, a meter of cable costs $\$12$ and you need a piece of $350\text{ cm}$. How do you know how much to pay if the unit of the price and the unit of your amount are not the same?
The price per unit
For an Art project, Marcos needs to buy $2 \text{ m}$ and $40 \text{ cm}$ of decorative ribbon. If a meter of ribbon costs $\$10…
For an Art project, Marcos needs to buy $2 \text{ m}$ and $40 \text{ cm}$ of decorative ribbon. If a meter of ribbon costs $\$10$, what is the total cost of the ribbon Marcos needs?
The key step: convert before multiplying
Almost all mistakes come from multiplying without first converting the amount to the same unit as the price. Remember the most commonly used equivalences:
Mass and capacity work the same way (g, dg, cg, mg; L, dL, cL, mL). In addition:
| Equivalence | Value |
|---|---|
| $1\text{ t}$ | $1{,}000\text{ kg}$ |
| $1\text{ kg}$ | $1{,}000\text{ g}$ |
| $1\text{ dg}$ | $0.1\text{ g}$ |
| $1\text{ daL}$ | $10\text{ L}$ |
| $1\text{ cL}$ | $0.01\text{ L}$ |
| $1\text{ ha}$ | $10{,}000\text{ m}^2$ |
| $1\text{ m}^3$ | $1{,}000\text{ L}$ |
A delivery van can transport a maximum of $1.5 \text{ t}$ (metric tons). If the transport cost is $\$2…
A delivery van can transport a maximum of $1.5 \text{ t}$ (metric tons). If the transport cost is $\$2$ per kilogram of load, how much will the delivery cost if the van is filled to its maximum capacity?
In a stationer's, a gram of special craft paper costs $\$0.40$. If we buy a sheet weighing $80 \text{ dg}$, how much will we have to pay?
In a stationer's, a gram of special craft paper costs $\$0.40$. If we buy a sheet weighing $80 \text{ dg}$, how much will we have to pay?
In your local greengrocer's, a kilogram of Jerte cherries costs $\$5.80$. If you decide to buy a bag containing $750 \text{ g}$ of cherries and pay with a $\$10$ note,…
In your local greengrocer's, a kilogram of Jerte cherries costs $\$5.80$. If you decide to buy a bag containing $750 \text{ g}$ of cherries and pay with a $\$10$ note, how much change will you receive?
Prices with lengths and areas
The same method works when the price is per meter or per square meter: you convert the measurements to meters and, if necessary, first calculate the area.
Lucía's mother has bought a rectangular rug for her bedroom measuring $3 \text{ m}$ long and $2 \text{ m}$ wide. If the price of the rug is $\$300…
Lucía's mother has bought a rectangular rug for her bedroom measuring $3 \text{ m}$ long and $2 \text{ m}$ wide. If the price of the rug is $\$300$ per square meter ($m^2$), how much did the rug cost?
You want to replace the floor in your bedroom, which is rectangular with dimensions of $4 \text{ m}$ long and $350 \text{ cm}$ wide. The price of the laminate flooring…
You want to replace the floor in your bedroom, which is rectangular with dimensions of $4 \text{ m}$ long and $350 \text{ cm}$ wide. The price of the laminate flooring you like is $\$370$ per square meter. What will be the total cost of the material to cover the whole floor?
A farmer wants to fence off a rectangular field for his sheep measuring $1.5 \text{ hm}$ long and $80 \text{ m}$ wide. A meter of metal fencing costs $\$128…
A farmer wants to fence off a rectangular field for his sheep measuring $1.5 \text{ hm}$ long and $80 \text{ m}$ wide. A meter of metal fencing costs $\$128$. Furthermore, 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 required fencing?
Capacities, volumes, and density
A liquid is sold in liters, but a tank is measured in $\text{m}^3$. The key: $1\text{ dm}^3 = 1\text{ L}$, and therefore $1\text{ m}^3 = 1{,}000\text{ L}$. In the same way, $1\text{ cm}^3 = 1\text{ mL}$.
To go from volume to mass you use density: how many kilograms or grams there are in each unit of volume.
A tanker lorry carries $15 \text{ m}^3$ of drinking water to supply a drought-affected area. a) If water is sold at $\$1$ per decaliter ($\text{daL}…
A tanker lorry carries $15 \text{ m}^3$ of drinking water to supply a drought-affected area.
- a) If water is sold at $\$1$ per decaliter ($\text{daL}$), how much is the lorry's entire load worth?
- b) How many $5 \text{ L}$ water containers could be filled with that entire volume?
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 $\$2\,900$ per kilogram. In addition, the glass bottle costs $\$43$ and the packaging $\$16$. If they want to make a profit of $\$300$ per bottle sold, what should the retail price of a single bottle be?
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 packaged into $75 \, cl$…
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 packaged into $75 \, cl$ bottles and each bottle is sold on the international market at a price of $12.50$ dollars, what is the total value of the tank in dollars? (Assume that the tank is completely emptied).
Comparing deals
To find out which is the better deal, calculate the price of the same amount in each offer, for example per kilogram or per liter.
In a supermarket, we find two offers for jars of jam of the same brand: Option A: A $450 \, g$ jar costing $\$2.85$. Option B: A $0.6 \, kg$ jar with an offer of "$20 \,…
In a supermarket, we find two offers for jars of jam of the same brand:
-
Option A: A $450 \, g$ jar costing $\$2.85$.
-
Option B: A $0.6 \, kg$ jar with an offer of "$20 \, \%$ extra free" (that is, 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 one.
Artificial turf is to be installed on a training field that measures $0.4 \, hm$ long and $250 \, dm$ wide. There are two quotes: Company "VerdeTotal": $\$370…
Artificial turf is to be installed on a training field that measures $0.4 \, hm$ long and $250 \, dm$ wide. There are two quotes:
-
Company "VerdeTotal": $\$370$ per square meter, but they charge a fixed installation fee of $\$9\,000$.
-
Company "PraderaPlus": $\$0.04$ 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?
Problems with several pieces
Some problems mix amounts in different units or have several steps: convert them all, add, and finally multiply by the price.
In a hardware store, network cable is sold at $\$17$ 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 $\$17$ 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 gives you a discount of $\$50$ for buying more than $20 \text{ m}$ in total, how much will you have to pay in the end?
An organic farmer has a citrus orchard covering an area of $0.85 \, ha$. To keep the trees hydrated during the summer, he uses an irrigation system that consumes exactly…
An organic farmer has a citrus orchard covering an area of $0.85 \, ha$. To keep the trees hydrated during the summer, he uses an irrigation system that consumes exactly $12 \, L$ of water per square meter per day. If the price of irrigation water is $\$7$ per cubic meter ($m^3$), how much will the water bill come to after a month of $30$ days?
We want to paint the four walls and ceiling of an exhibition hall shaped like a rectangular prism. The dimensions of the floor plan are $12 \, m$ long by $80 \, dm$…
We want to paint the four walls and ceiling of an exhibition hall shaped like a rectangular prism. The dimensions of the floor plan 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 window wall that occupies $15 \, m^2$.
-
The coverage of the paint is $1 \, L$ per $10 \, m^2$.
-
The paint is sold in cans of $5 \, L$ at a price of $\$850$ per can.
Calculate the total cost of the required paint, taking into account that only full cans can be bought.
Go back to the beginning: the $750\text{ g}$ of tomatoes are $0.75\text{ kg}$, which at $\$28$ per kilogram cost $\$21$; and the $350\text{ cm}$ of cable are $3.5\text{ m}$, which at $\$12$ per meter cost $\$42$. Always: convert and then multiply.