Mass
At the tortillería they ask you for "a kilo," the sugar bag at the supermarket says "500 g," and a delivery truck can't carry more than a few metric tons. Your backpack, in addition, shouldn't weigh more than 10% of what you weigh. All of this is measured with the same family of units: the gram and its multiples and submultiples. How much can your backpack weigh without hurting your back? We'll solve it at the end.
The unit of mass: the gram
In daily life we say "weigh" to mean "measure the mass." To be precise, mass is the amount of matter in a body, and the main unit is the gram ($\text{g}$).
A quantity is what we measure (mass), the unit is what we express it in (gram, kilogram), and the instrument is what we use to measure (balance, scale).
In the study of matter, it is essential not to confuse what we measure with the tool we use. Classify the following terms in the table according to whether they…
In the study of matter, it is essential not to confuse what we measure with the tool we use. Classify the following terms in the table according to whether they correspond to a quantity, a unit of measurement or a measuring instrument:
Mass, Gram, Balance, Kilogram, Scales, Capacity, Metric ton, Dynamometer.
It is very important to know how to choose the appropriate unit depending on the object we are weighing. Match each of the following objects with the measurement that…
It is very important to know how to choose the appropriate unit depending on the object we are weighing. Match each of the following objects with the measurement that seems most realistic:
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A worker ant.
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A full can of drink.
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A laptop.
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A delivery lorry.
Possible measurements: $2\text{ kg}$, $8\text{ t}$, $3\text{ mg}$, $330\text{ g}$.
Gram, kilo, and metric ton
This is a review from elementary school. The three units you use most are:
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the gram ($\text{g}$), for small things;
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the kilogram ($\text{kg}$), the "kilo" at the tortillería: $1\ \text{kg} = 1{,}000\ \text{g}$;
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the metric ton ($\text{t}$), for big loads: $1\ \text{t} = 1{,}000\ \text{kg}$.
For tiny things the milligram ($\text{mg}$) is used: $1\ \text{g} = 1{,}000\ \text{mg}$.
Some problems use other names. Each unit in the table is 10 times the one that follows, and the last two rows are equivalences worth remembering:
| Unit | Symbol | Equivalence |
|---|---|---|
| kilogram | $\text{kg}$ | $1{,}000\ \text{g}$ |
| hectogram | $\text{hg}$ | $100\ \text{g}$ |
| decagram | $\text{dag}$ | $10\ \text{g}$ |
| gram | $\text{g}$ | $1\ \text{g}$ |
| decigram | $\text{dg}$ | $0.1\ \text{g}$ |
| centigram | $\text{cg}$ | $0.01\ \text{g}$ |
| milligram | $\text{mg}$ | $0.001\ \text{g}$ |
| metric quintal | $\text{q}$ | $100\ \text{kg}$ |
| metric ton | $\text{t}$ | $1{,}000\ \text{kg}$ |
A van has a maximum load capacity of $1.5$ metric metric tons ($t$). If we want to load boxes of oranges weighing $25\text{ kg}$ each, what is the maximum number of…
A van has a maximum load capacity of $1.5$ metric metric tons ($t$). If we want to load boxes of oranges weighing $25\text{ kg}$ each, what is the maximum number of boxes we can transport in a single trip? (Remember that $1\text{ t} = 1,000\text{ kg}$).
To make a chocolate cake for her sister's birthday, Ainhoa bought the following ingredients: A packet of flour of $0.75\text{ kg}$. A bar of chocolate of $200\text{ g}$.…
To make a chocolate cake for her sister's birthday, Ainhoa bought the following ingredients:
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A packet of flour of $0.75\text{ kg}$.
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A bar of chocolate of $200\text{ g}$.
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A packet of sugar of $4.5\text{ hg}$.
What is the total mass of the ingredients in grams?
In the school vegetable garden, $4.2$ metric tons of apples have been harvested. For transport, sacks containing one metric quintal ($q$) each are used. a) How many…
In the school vegetable garden, $4.2$ metric tons of apples have been harvested. For transport, sacks containing one metric quintal ($q$) each are used.
- a) How many sacks are needed to transport the entire harvest?
- b) If $1\ kg$ of apple compote is produced from every $5\ kg$ of apples, how many kilograms of compote will the school produce from that harvest?
Changing units
To go to a smaller unit, the number grows: you multiply. To go to a larger one, the number decreases: you divide. Since the system is decimal, multiplying or dividing by $10$, $100$, $1{,}000$… just moves the decimal point. With the table from the previous section you know by how much.
To convert from one unit to another in the metric system, we must multiply or divide by 1 followed by zeros. Complete the following equalities by converting to the…
To convert from one unit to another in the metric system, we must multiply or divide by 1 followed by zeros. Complete the following equalities by converting to the indicated unit:
- a) $45\text{ kg} = \dots\dots\dots\text{ g}$
- b) $1,200\text{ mg} = \dots\dots\dots\text{ g}$
- c) $0.5\text{ hg} = \dots\dots\dots\text{ dag}$
- d) $34.7\text{ dg} = \dots\dots\dots\text{ cg}$
In a science laboratory, three students have weighed the same rock sample. Each has written down the result in a different unit: Student A: $0.45\text{ kg}$ Student B:…
In a science laboratory, three students have weighed the same rock sample. Each has written down the result in a different unit:
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Student A: $0.45\text{ kg}$
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Student B: $4,200\text{ dg}$
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Student C: $46\text{ dag}$
Express all measurements in grams and order the students from lowest to highest according to the mass they obtained.
A well-known bakery in your neighbourhood uses the following mass amounts to make its famous "Imperial Cake": Flour: $0.55\ kg$ Sugar: $22.5\ dag$ Butter: $1,250\ dg$…
A well-known bakery in your neighbourhood uses the following mass amounts to make its famous "Imperial Cake":
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Flour: $0.55\ kg$
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Sugar: $22.5\ dag$
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Butter: $1,250\ dg$
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Chocolate: $3\ hg$
Express each of the ingredients in grams and calculate the total mass of the cake (without including the filling or decoration) in compound form (using kilograms and grams).
Complex form, incomplex form, and comparing masses
A mass in complex form uses several units ($3\ \text{kg}$ and $450\ \text{g}$). In incomplex form it uses a single one ($3{,}450\ \text{g}$). To compare or add masses, first convert them to the same unit.
Express the following quantities in simple form (using a single unit), converting them to the unit indicated in each case: a) $3\text{ kg y } 450\text{ g}$ $\rightarrow$…
Express the following quantities in simple form (using a single unit), converting them to the unit indicated in each case:
- a) $3\text{ kg y } 450\text{ g}$ $\rightarrow$ into grams.
- b) $5\text{ dag, } 2\text{ g y } 8\text{ dg}$ $\rightarrow$ into decigrams.
- c) $1\text{ t y } 200\text{ kg}$ $\rightarrow$ into kilograms.
- d) $4\text{ hg y } 5\text{ cg}$ $\rightarrow$ into centigrams.
Order the following masses of electronic components used in manufacturing a smartphone from smallest to largest. To compare them, convert all units to milligrams ($mg$):…
Order the following masses of electronic components used in manufacturing a smartphone from smallest to largest. To compare them, convert all units to milligrams ($mg$):
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Camera sensor: $0.0042\ kg$
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Main microchip: $1.85\ g$
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Fastening screw: $85\ cg$
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Charging connector: $12.4\ dg$
A balance scale is in equilibrium. On the left pan there is a full jar of honey weighing $1.25$ $kg$. On the right pan there is a $5$ $hg$ weight, a $25$ $dag$ weight…
A balance scale is in equilibrium. On the left pan there is a full jar of honey weighing $1.25$ $kg$. On the right pan there is a $5$ $hg$ weight, a $25$ $dag$ weight and the same jar of honey, but empty. Calculate the mass of the honey inside the jar and the mass of the empty jar separately, expressing both results in grams.
Other units: myriagram and microgram
Some problems use uncommon units. They are handled the same way: you just need to know their equivalence.
| Unit | Symbol | Equivalence |
|---|---|---|
| myriagram | $\text{mag}$ | $10\ \text{kg}$ |
| microgram | $\mu\text{g}$ | $1\ \text{mg} = 1{,}000\ \mu\text{g}$ |
A pharmaceutical company manufactures vitamin supplement capsules. Each capsule contains $150$ $mg$ of active ingredient, $0.025$ $dg$ of excipient and $4,500$ $\mu g$…
A pharmaceutical company manufactures vitamin supplement capsules. Each capsule contains $150$ $mg$ of active ingredient, $0.025$ $dg$ of excipient and $4,500$ $\mu g$ (micrograms) of gel coating. Knowing that $1$ $mg = 1,000$ $\mu g$, calculate the total mass of a tub containing $120$ capsules, expressing the final result in grams. Do not forget to add the $12$ $g$ that the empty plastic tub weighs.
To carry out a chemistry experiment, a student needs to mix three substances. He has $0.045$ $kg$ of substance A, $12.4$ $dag$ of substance B and $7,800$ $cg$ of…
To carry out a chemistry experiment, a student needs to mix three substances. He has $0.045$ $kg$ of substance A, $12.4$ $dag$ of substance B and $7,800$ $cg$ of substance C. After mixing them, a reaction takes place that releases a gas, causing the mixture to lose one hundredth of its initial total mass. What will be the final mass of the resulting liquid mixture expressed in grams?
A bulk food shop receives an order of rice in myriagram bags ($1$ $mag = 10$ $kg$). The shop owner wants to repackage the rice into smaller bags of three different…
A bulk food shop receives an order of rice in myriagram bags ($1$ $mag = 10$ $kg$). The shop owner wants to repackage the rice into smaller bags of three different sizes:
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Half of the rice will be packed into $500$ $g$ bags.
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Two fifths of the remaining rice will be packed into $250$ $g$ bags.
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Whatever is left will be packed into $100$ $g$ bags.
If they received $12$ myriagram bags in total, how many bags of each type will they get?
Water and mass: liters and kilos
A very useful fact: a liter of water has a mass of $1\ \text{kg}$, and a cubic decimeter ($1\ \text{dm}^3$) is the same as a liter. Other materials weigh differently per liter; the next section is about that.
Did you know that the kilogram was originally defined based on water? One liter of pure water has a mass of exactly one kilogram. If you have a backpack that can hold a…
Did you know that the kilogram was originally defined based on water? One liter of pure water has a mass of exactly one kilogram. If you have a backpack that can hold a maximum weight of $6\text{ kg}$ and you already have books inside that weigh $2500\text{ g}$, how many $1.5$-liter bottles of water can you add at most without the backpack breaking?
Knowing that one cubic decimeter ($1\ dm^3$) of distilled water has a mass of exactly $1\ kg$, solve the following problem: An aquarium is shaped like a rectangular…
Knowing that one cubic decimeter ($1\ dm^3$) of distilled water has a mass of exactly $1\ kg$, solve the following problem:
An aquarium is shaped like a rectangular prism with dimensions of $60\ cm$ long, $30\ cm$ wide and $40\ cm$ high. If it is filled with water up to $\frac{3}{4}$ of its total capacity, what will be the mass of the water inside the aquarium in kilograms?
Mass per cm³
A liter of water weighs $1\ \text{kg}$, but a liter of oil weighs less and one of lead, a lot more. To compare materials we use the mass that each $\text{cm}^3$ has: it is a unit value, like the price of a kilo.
Water has $1\ \text{g/cm}^3$, and that is why $1{,}000\ \text{cm}^3$ (a liter) weigh $1{,}000\ \text{g} = 1\ \text{kg}$. Oil has $0.92\ \text{g/cm}^3$: it weighs less than water and floats on it.
The mass of a solid metal cube with an edge length of $5$ $cm$ was measured and found to be $343.75$ $g$. What is the mass of this metal per cubic centimeter ($cm^3$)?…
The mass of a solid metal cube with an edge length of $5$ $cm$ was measured and found to be $343.75$ $g$.
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What is the mass of this metal per cubic centimeter ($cm^3$)?
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If we have another block of the same metal with a volume of $1.2$ $dm^3$, what will its mass be in kilograms?
(Note: The volume of a cube is calculated as $V = arista \times arista \times arista$)
In an oil factory, a bottling machine fills $0.75$-liter bottles. We know that $1$ liter of this oil has a mass of $920$ $g$. If a lorry transports $150$ boxes and each…
In an oil factory, a bottling machine fills $0.75$-liter bottles. We know that $1$ liter of this oil has a mass of $920$ $g$. If a lorry transports $150$ boxes and each box contains $12$ bottles, what is the total mass of the oil cargo in metric tons? If each empty bottle has a mass of $35$ $dag$ and each cardboard box weighs $0.4$ $kg$, also calculate the total mass of the shipment (oil + bottles + boxes) in kilograms.
Price by weight and sharing
When a price is given per unit of mass, first convert the amount you buy to that same unit and then multiply. For the reverse path, divide.
In a candy shop that sells sweets by weight, the price of marshmallows is $\$1.20$ per hectogram ($hg$). a) How much will a bag containing $250\ g…
In a candy shop that sells sweets by weight, the price of marshmallows is $\$1.20$ per hectogram ($hg$).
- a) How much will a bag containing $250\ g$ of marshmallows cost?
- b) If we paid $\$6$ for another bag of marshmallows, how many grams of marshmallows did we buy?
The robotics club has an aluminum bar of $1.2$ $kg$. They want to make as many pieces as possible of $12$ $g$ and $500$ $mg$ each. What is the maximum number of pieces…
The robotics club has an aluminum bar of $1.2$ $kg$. They want to make as many pieces as possible of $12$ $g$ and $500$ $mg$ each.
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What is the maximum number of pieces they can make?
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If one gram of aluminum costs $70$ pesos, how much will the material for all the manufactured pieces cost?
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How much aluminum will be left over in milligrams?
A delivery lorry has a maximum load capacity of $3.5$ metric metric tons. At the warehouse, $12$ boxes of machinery weighing $1.45$ metric quintals each and $45$ bags of…
A delivery lorry has a maximum load capacity of $3.5$ metric metric tons. At the warehouse, $12$ boxes of machinery weighing $1.45$ metric quintals each and $45$ bags of animal feed weighing $25$ $kg$ each have been loaded. If a container of metal parts with a mass of $875,000$ grams still needs to be loaded, will the lorry be able to carry all the cargo in one go without exceeding its limit? Justify your answer by calculating how many kilograms below or over the maximum capacity it is.
Problems with masses
To solve a problem with masses, follow three steps: convert everything to the same unit, operate, and give the answer in the unit that is asked for.
Go back to the beginning: the backpack of a person who weighs $45\ \text{kg}$ shouldn't exceed $4.5\ \text{kg}$. To find that out, it is enough to calculate the $10\ \%$ and compare in the same unit.
To prevent back injuries, it is recommended that a school backpack does not exceed $10 \%$ of the student's weight. Lucas weighs $45\ kg$ and has weighed the items he is…
To prevent back injuries, it is recommended that a school backpack does not exceed $10 \%$ of the student's weight. Lucas weighs $45\ kg$ and has weighed the items he is carrying to class today:
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Books and notebooks: $2.8\ kg$
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Pencil case with stationery: $350\ g$
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English dictionary: $6.5\ hg$
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Full water bottle: $75\ dag$
Calculate the total mass of the backpack in kilograms and determine whether Lucas is carrying a safe weight for his back.
A delivery lorry has a maximum authorised load of $3.5\ t$. It is currently carrying $18$ pallets of milk boxes weighing $125\ kg$ each. The driver needs to pick up $12$…
A delivery lorry has a maximum authorised load of $3.5\ t$. It is currently carrying $18$ pallets of milk boxes weighing $125\ kg$ each. The driver needs to pick up $12$ bags of animal feed weighing $450\ hg$ each at a warehouse.
Will he be able to load the bags of animal feed without exceeding the lorry's mass limit? Justify your answer by performing the calculations in kilograms.
A pharmacy receives a delivery of $5$ boxes of a medicine. Each box contains $12$ tubs, and each tub weighs $150\ g$ when empty. Inside each tub there are $60$ capsules,…
A pharmacy receives a delivery of $5$ boxes of a medicine. Each box contains $12$ tubs, and each tub weighs $150\ g$ when empty. Inside each tub there are $60$ capsules, and each capsule has a mass of $500\ mg$.
Calculate the total mass of the delivery (including tubs and capsules) expressed in kilograms.