Price Problems
At the corner colmado, a pound of cheese costs a fixed price, and so does a meter of cable at the hardware store. When you know the price of one unit, you can calculate the price of any amount... as long as you first convert it to that unit. How do you avoid mistakes when doing this, and how do you know which offer is the better deal?
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 $\text{RD\$}\ 30…
For an Art project, Marcos needs to buy $2 \text{ m}$ and $40 \text{ cm}$ of decorative ribbon. If a meter of ribbon costs $\text{RD\$}\ 30$, what is the total cost of the ribbon Marcos needs?
The key step: convert before multiplying
Almost all mistakes in these problems come from multiplying without having converted the quantity to the same unit as the price.
Other useful equivalences: $1\text{ t} = 1{,}000\text{ kg}$ and $1\text{ dg} = 0.1\text{ g}$.
In a stationer's, a gram of special craft paper costs $\text{RD\$}\ 1.20$. 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 $\text{RD\$}\ 1.20$. If we buy a sheet weighing $80 \text{ dg}$, how much will we have to pay?
A delivery van can transport a maximum of $1.5 \text{ t}$ (metric tons). If the transport cost is $\text{RD\$}\ 6…
A delivery van can transport a maximum of $1.5 \text{ t}$ (metric tons). If the transport cost is $\text{RD\$}\ 6$ per kilogram of load, how much will the delivery cost if the van is filled to its maximum capacity?
In your local greengrocer's, a kilogram of Jerte cherries costs $\text{RD\$}\ 5.80$. If you decide to buy a bag containing $750 \text{ g}…
In your local greengrocer's, a kilogram of Jerte cherries costs $\text{RD\$}\ 5.80$. If you decide to buy a bag containing $750 \text{ g}$ of cherries and pay with a $\text{RD\$}\ 10$ 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 quantity to that unit and multiply.
When the price is per square centimeter
Area units change by factors of 100 at each step, so going from $\text{cm}^2$ to $\text{m}^2$ takes two steps: $1\text{ m}^2 = 100 \times 100 = 10{,}000\text{ cm}^2$.
If a price is given per $\text{cm}^2$, you can convert it to a price per $\text{m}^2$ by multiplying it by $10{,}000$. Another way is to convert the area to $\text{cm}^2$ and multiply by the price per $\text{cm}^2$.
Walls and ceiling of a room
A room shaped like a rectangular prism (box) has four walls and a ceiling. The walls make up the lateral area: it is the perimeter of the base times the height. Adding the ceiling, you get the area that gets painted.
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 $\text{RD\$}\ 900…
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 $\text{RD\$}\ 900$ 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 $\text{RD\$}\ 1\,110$ 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 $\text{RD\$}\ 384…
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 $\text{RD\$}\ 384$. 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?
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:
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One of the long walls has a window wall that occupies $15 \, m^2$.
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The coverage of the paint is $1 \, L$ per $10 \, m^2$.
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The paint is sold in cans of $5 \, L$ at a price of $\text{RD\$}\ 2\,550$ per can.
Calculate the total cost of the required paint, taking into account that only full cans can be bought.
Comparing offers
To decide which offer is the better deal, calculate the price of the same quantity in both (for example, per kilogram or per liter).
For a birthday party, you need to buy $12 \text{ L}$ of orange drink. At the supermarket you have two options: $1.5 \text{ L}$ bottles at $\text{RD\$}\ 72…
For a birthday party, you need to buy $12 \text{ L}$ of orange drink. At the supermarket you have two options:
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$1.5 \text{ L}$ bottles at $\text{RD\$}\ 72$ each.
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Packs of 6 cans of $33 \text{ cl}$ at $\text{RD\$}\ 189$ per pack.
Calculate how much it would cost you to buy the exact $12 \text{ L}$ using only the first option. How many packs of cans would you have to buy to have at least that amount?
In a supermarket, we find two offers for jars of jam of the same brand: Option A: A $450 \, g$ jar costing $\text{RD\$}\ 2.85$. Option B: A $0.6 \, kg…
In a supermarket, we find two offers for jars of jam of the same brand:
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Option A: A $450 \, g$ jar costing $\text{RD\$}\ 2.85$.
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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 $\text{RD\$}\ 3.90$.
Determine which of the two options is cheaper by calculating the price per kilogram of each one.
Problems with several sections and several steps
Some problems mix quantities in different units or ask for several calculations: convert everything to the same unit, add, and finally multiply by the price.
If the problem uses volumes or density, first convert everything to a common unit: $1\text{ dm}^3 = 1\text{ L}$, $1\text{ m}^3 = 1{,}000\text{ L}$, $1\text{ cm}^3 = 1\text{ mL}$, $1\text{ cL} = 0.01\text{ L}$, $1\text{ daL} = 10\text{ L}$, and $1\text{ ha} = 10{,}000\text{ m}^2$. If you have to buy whole containers or cans, round the number of units up.
Volume of a cube and of a rectangular prism
To find out how much liquid fits in a tank, you calculate its volume. For a rectangular prism (box):
$$V = \text{length} \times \text{width} \times \text{height}$$In a cube all three measurements are equal to the edge $a$: $V = a \times a \times a$.
A tanker lorry carries $15 \text{ m}^3$ of drinking water to supply a drought-affected area. a) If water is sold at $\text{RD\$}\ 3$ 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 $\text{RD\$}\ 3$ 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?
In a hardware store, network cable is sold at $\text{RD\$}\ 51$ per meter. You need to buy several pieces to install internet at home: one of $1250 \text{ cm}…
In a hardware store, network cable is sold at $\text{RD\$}\ 51$ 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 $\text{RD\$}\ 150$ 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 $\text{RD\$}\ 0.35$ per cubic meter ($m^3$), how much will the water bill come to after a month of $30$ days?
Density, profit, and budgets
In the longer problems, several steps are chained together. Density tells you how many grams each $\text{cm}^3$ has: $\text{mass} = \text{density} \times \text{volume}$. Remember that $1\text{ mL} = 1\text{ cm}^3$ and that $1\text{ dm}^3 = 1\text{ L}$.
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 $\text{RD\$}\ 8\,700$ per kilogram. In addition, the glass bottle costs $\text{RD\$}\ 129$ and the packaging $\text{RD\$}\ 48$. If they want to make a profit of $\text{RD\$}\ 900$ 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).
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": $\text{RD\$}\…
Artificial turf is to be installed on a training field that measures $0.4 \, hm$ long and $250 \, dm$ wide. There are two quotes:
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Company "VerdeTotal": $\text{RD\$}\ 1\,110$ per square meter, but they charge a fixed installation fee of $\text{RD\$}\ 27\,000$.
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Company "PraderaPlus": $\text{RD\$}\ 0.13$ 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?
Back to the beginning: to find the price of any quantity, convert to the unit of the price and multiply; to compare offers, calculate the price per the same unit.