Metric Units of Mass
Your backpack shouldn't weigh more than about 10% of your body weight. The nutrition label on a granola bar says "Sodium 95 mg." A shipping container leaving a port has its load limit printed in metric tons. All three are measuring the same thing, mass, but in very different sizes of units. How do you move between them without getting lost?
Mass and the Gram
The amount of matter in an object is its mass. In everyday talk we say "weight," and in this chapter "weigh" just means "find the mass." In the US you use pounds and ounces at the grocery store, but nutrition labels, medicine, science class, and international shipping all use the metric system for mass.
It helps to keep three ideas apart: what you measure (the magnitude), how much you count it in (the unit), and what you use to measure it (the tool).
| Magnitude | Unit | Tool |
|---|---|---|
| length | inch, meter | ruler |
| time | second | stopwatch |
| temperature | °F | thermometer |
| mass | gram, kilogram | 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 magnitude, a unit of measurement, or a measuring instrument:
Mass, Gram, Balance, Kilogram, Scale, Capacity, Metric ton, Dynamometer.
The Metric Ladder for Mass
Metric units come in families. Each unit is 10 times the next smaller one, because the system is built on place value, just like ten tens make a hundred.
| Unit | Symbol | Mass in grams |
|---|---|---|
| kilogram | kg | 1,000 g |
| hectogram | hg | 100 g |
| decagram | dag | 10 g |
| gram | g | 1 g |
| decigram | dg | 0.1 g |
| centigram | cg | 0.01 g |
| milligram | mg | 0.001 g |
| microgram | µg | 0.000001 g |
In everyday US life you will mostly see kg, g, and mg (and µg on vitamin labels). The units hg, dag, dg, and cg are rarely used outside of classrooms. They appear in practice problems so that you learn the pattern of the whole ladder.
For big loads there are three more units. They are bigger than the kilogram, and you will meet them mostly in problems about international trade and freight:
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The metric ton ($\text{t}$): $1\text{ t} = 1{,}000\text{ kg}$.
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The metric quintal ($\text{q}$): $1\text{ q} = 100\text{ kg}$, so $1\text{ t} = 10\text{ q}$.
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The myriagram ($\text{mag}$): $1\text{ mag} = 10\text{ kg}$.
The microgram is much smaller than the milligram: $1\text{ mg} = 1{,}000\ \mu\text{g}$.
To choose a sensible unit, it helps to know some benchmarks. Roughly, $1\text{ kg} \approx 2.2\text{ lb}$ and $1\text{ oz} \approx 28\text{ g}$.
| Object | Metric mass | Customary comparison |
|---|---|---|
| grain of rice | 20 mg | too light to feel |
| paper clip | 1 g | about $\tfrac{1}{28}$ of an ounce |
| nickel | 5 g | about $\tfrac{1}{6}$ of an ounce |
| math textbook | 1 kg | a little over 2 lb |
| gallon of milk | 4 kg | about 8.6 lb |
| small car | 1.5 t | about 3,300 lb |
It is very important to know how to choose the appropriate unit according to 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 according to the object we are weighing. Match each of the following objects with the measurement that seems most realistic to you:
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A worker ant.
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A full can of soda.
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A laptop.
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A delivery truck.
Possible measurements: $2\text{ kg}$, $8\text{ t}$, $3\text{ mg}$, $330\text{ g}$.
Converting Units
To change units, count the steps on the ladder. Each step moves the decimal point one place.
Units like the metric ton work the same way, just with a bigger jump: $3.4\text{ t} = 3.4 \times 1{,}000 = 3{,}400\text{ kg}$, and $750\text{ kg} = 750 \div 1{,}000 = 0.75\text{ t}$.
To convert from one unit to another in the metric system, we must multiply or divide by one followed by zeros. Complete the following equalities by performing the…
To convert from one unit to another in the metric system, we must multiply or divide by one followed by zeros. Complete the following equalities by performing the indicated unit conversions:
- 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 recorded 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 recorded 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 the measurements in grams and order the students from smallest to largest according to the mass they have obtained.
A well-known bakery in your neighborhood uses the following quantities of mass to make its famous "Imperial Cake": Flour: $0,55\ kg$ Sugar: $22,5\ dag$ Butter:…
A well-known bakery in your neighborhood uses the following quantities of mass 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 (excluding the filling and decoration) in complex form (using kilograms and grams).
Adding and Comparing Masses
You can only add or compare masses when they are in the same unit. $2\text{ kg} + 300\text{ g}$ is not $302$ of anything.
Sometimes a mass is given in several units at once, like $3\text{ kg } 450\text{ g}$ (mixed units). Sometimes it is written in a single unit, like $3{,}450\text{ g}$. Some problems call these complex form and incomplex form. To switch to a single unit, convert each part and add: $3\text{ hg } 4\text{ dag } 6\text{ g} = 300 + 40 + 6 = 346\text{ g}$. To go back to mixed units, divide: $4{,}275\text{ g} = 4\text{ kg } 275\text{ g}$.
The number line below shows the three masses in milligrams: $2{,}500$, $3{,}100$, and $3{,}400$. The farther right, the greater the mass.
To prepare a chocolate cake for her sister's birthday, Lucy has bought the following ingredients: A packet of flour of $0,75\text{ kg}$. A bar of chocolate of…
To prepare a chocolate cake for her sister's birthday, Lucy has 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?
Express the following quantities in incomplex form (using a single unit), converting them to the unit indicated in each case: a) $3\text{ kg y } 450\text{ g}$…
Express the following quantities in incomplex form (using a single unit), converting them to the unit indicated in each case:
- a) $3\text{ kg y } 450\text{ g}$ $\rightarrow$ in grams.
- b) $5\text{ dag, } 2\text{ g y } 8\text{ dg}$ $\rightarrow$ in decigrams.
- c) $1\text{ t y } 200\text{ kg}$ $\rightarrow$ in kilograms.
- d) $4\text{ hg y } 5\text{ cg}$ $\rightarrow$ in centigrams.
Order the following masses of electronic components used in the manufacture of a smartphone from smallest to largest. To compare, convert all units to milligrams ($mg$):…
Order the following masses of electronic components used in the manufacture of a smartphone from smallest to largest. To compare, 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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Fixing screw: $85\ cg$
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Charging connector: $12,4\ dg$
Water and Mass
Water links volume and mass in a very handy way. This is not an accident: the metric system was built to match.
Other materials are not like water. Their mass for each $\text{cm}^3$ is different, but it is a unit rate you can find.
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 weighing $2.500\text{ g}$, what is the maximum number of $1,5$-liter water bottles you can add without breaking the backpack?
Knowing that one cubic decimeter ($1\ dm^3$) of distilled water has a mass of exactly $1\ kg$, solve the following problem: An aquarium has the shape of 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 has the shape of 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 contained in the aquarium expressed in kilograms?
The mass of a solid metal cube with an edge of $5$ $cm$ has been measured and a value of $343,75$ $g$ has been obtained. What is the mass of this metal per cubic…
The mass of a solid metal cube with an edge of $5$ $cm$ has been measured and a value of $343,75$ $g$ has been obtained.
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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 expressed in kilograms?
(Fact: The volume of a cube is calculated as $V = arista \cdot arista \cdot arista$)
Totals and Limits
Many real problems ask: "Is the total under the limit?" The plan is always the same: pick one unit, find the total, compare with the limit.
Health class often uses the metric system, so let's say Maya's mass is $50\text{ kg}$ (about $110\text{ lb}$). The "10% of body weight" backpack rule works in any unit.
Freight shipped between countries is measured in metric tons, so load limits for cargo vans and trucks are a good place to practice.
When the question is "how many fit?", divide the limit by the mass of one item. Only whole items count, so ignore any leftover. For example, $1{,}000\text{ kg} \div 90\text{ kg} = 11.1\ldots$ means at most $11$ sacks of $90\text{ kg}$.
A van has a maximum load capacity of $1,5$ metric tons ($t$). If we want to load boxes of oranges weighing $25\text{ kg}$ each, what is the maximum number of boxes we…
A van has a maximum load capacity of $1,5$ 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 avoid 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 avoid 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 taking to class today:
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Books and notebooks: $2,8\ kg$
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Pencil case with materials: $350\ g$
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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 if Lucas is carrying a safe weight for his back.
A delivery truck has a maximum authorized load of $3,5\ t$. It is currently loaded with $18$ pallets of milk boxes weighing $125\ kg$ each. The driver must collect $12$…
A delivery truck has a maximum authorized load of $3,5\ t$. It is currently loaded with $18$ pallets of milk boxes weighing $125\ kg$ each. The driver must collect $12$ sacks of animal feed weighing $450\ hg$ each from a warehouse.
Will he be able to load the sacks of feed without exceeding the truck's mass limit? Justify your answer by performing the calculations in kilograms.
Big Loads: Groups Inside Groups
Shipments are often boxes of packages of items. Work from the inside out: one box first, then all the boxes.
If a problem tells you that $1\text{ L}$ of a liquid has a mass of, say, $800\text{ g}$, then $3$ liters have a mass of $3 \times 800\text{ g}$. Find the number of liters first, then multiply.
A pharmacy receives a shipment of $5$ boxes of a medicine. Each box contains $12$ bottles, and each bottle has a weight of $150\ g$ when empty. Inside each bottle, there…
A pharmacy receives a shipment of $5$ boxes of a medicine. Each box contains $12$ bottles, and each bottle has a weight of $150\ g$ when empty. Inside each bottle, there are $60$ capsules, and each capsule has a mass of $500\ mg$.
Calculate the total mass of the shipment (including bottles and capsules) expressed in kilograms.
A delivery truck has a maximum load capacity of $3,5$ metric tons. In the warehouse, $12$ boxes of machinery weighing $1,45$ metric quintals each and $45$ sacks of…
A delivery truck has a maximum load capacity of $3,5$ metric tons. In the warehouse, $12$ boxes of machinery weighing $1,45$ metric quintals each and $45$ sacks of animal feed of $25$ $kg$ each have been loaded. If a container of metal parts with a mass of $875.000$ grams still needs to be loaded, will the truck be able to transport the entire load at once without exceeding its limit? Justify your answer by calculating how many kilograms are left over or missing to reach the maximum.
In an oil factory, a packaging machine fills bottles of $0,75$ liters. We know that $1$ liter of this oil has a mass of $920$ $g$. If a truck transports $150$ boxes and…
In an oil factory, a packaging machine fills bottles of $0,75$ liters. We know that $1$ liter of this oil has a mass of $920$ $g$. If a truck 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 transport (oil + containers + boxes) in kilograms.
Rates, Prices, and Cutting Up a Mass
A price or a rate comes with its own unit. At a US store you might see "dollars per ounce" or "dollars per pound." In metric problems it might be "dollars per hectogram" (per $100\text{ g}$). The method is the same: change the mass to that unit first, then multiply the unit price by the amount.
A rate like "$1\text{ kg}$ of oil from every $5\text{ kg}$ of olives" works the same way. Change both masses to one unit, then divide the total by $5$. For example, $6.5\text{ t}$ of apples fills sacks of $1\text{ q}$ ($100\text{ kg}$) each: $6{,}500 \div 100 = 65$ sacks.
When you cut a mass into equal pieces, convert both masses to the same unit, then divide. The quotient is the number of whole pieces, and the remainder is what is left over.
In a bulk candy shop, the price of marshmallows is $1,20\ \$$ per hectogram ($hg$). a) How much will a bag containing $250\ g…
In a bulk candy shop, 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 have paid $6\ \$$ for another bag of candy, how many grams of candy have we bought?
In an agricultural cooperative in California, $4,2$ metric tons of olives have been harvested. For their transport, sacks containing one metric quintal ($q$) each are…
In an agricultural cooperative in California, $4,2$ metric tons of olives have been harvested. For their 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 oil is obtained from every $5\ kg$ of olives, how many kilograms of oil will the cooperative produce with this harvest?
An artisan jeweler has a silver bar of $1,2$ $kg$. He wants to make the greatest possible number of medals of $12$ $g$ and $500$ $mg$ each. What is the maximum number of…
An artisan jeweler has a silver bar of $1,2$ $kg$. He wants to make the greatest possible number of medals of $12$ $g$ and $500$ $mg$ each.
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What is the maximum number of medals he will be able to make?
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If he decides to sell each medal at $\$3,50$ per gram of silver, how much money will he get from the sale of all the medals made?
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How much silver will he have left over in milligrams?
Multi-Step Problems
Hard problems chain the ideas together. Write down what you know, convert to one unit, and take one step at a time. Three patterns come up a lot: a part of a total that is lost, fractions of an amount, and balances.
To find one hundredth of a mass, divide by $100$, which moves the decimal point 2 places left. Losing one hundredth ($1\%$) of a mass means subtracting that piece from the total.
A two-pan balance is level when both pans hold the same mass. Whatever is the same on both sides, such as an identical jar, can be ignored.
In the balance widget below, the unknown $x$ is the mass of the empty jar. The equation is $x + 550 = 1{,}200$. Use it to see that both sides are level when $x = 650$.
Back to the start. Now you can check all three things from the beginning. A $50\text{ kg}$ student's backpack limit is $5\text{ kg}$. A $95\text{ mg}$ label means $0.095\text{ g}$. A $2\text{ t}$ load limit is $2{,}000\text{ kg}$.
To carry out a chemical experiment, a student needs to mix three substances. Of substance A he has $0,045$ $kg$, of substance B he has $12,4$ $dag$ and of substance C he…
To carry out a chemical experiment, a student needs to mix three substances. Of substance A he has $0,045$ $kg$, of substance B he has $12,4$ $dag$ and of substance C he has $7.800$ $cg$. After mixing them, a reaction occurs 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 pharmaceutical company manufactures capsules of a vitamin supplement. Each capsule contains $150$ $mg$ of active ingredient, $0,025$ $dg$ of excipient and $4.500$…
A pharmaceutical company manufactures capsules of a vitamin supplement. Each capsule contains $150$ $mg$ of active ingredient, $0,025$ $dg$ of excipient and $4.500$ $\mu g$ (micrograms) of gelatinous coating. Knowing that $1$ $mg = 1.000$ $\mu g$, calculate the total mass of a bottle containing $120$ capsules, expressing the final result in grams. Do not forget to add the $12$ $g$ that the empty plastic bottle weighs.
A bulk food shop receives an order of rice in sacks of one myriagram ($1$ $mag = 10$ $kg$). The shop owner wants to repackage the rice into smaller bags of three…
A bulk food shop receives an order of rice in sacks of one myriagram ($1$ $mag = 10$ $kg$). 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 packaged in bags of $500$ $g$.
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Two-fifths of the remaining rice will be packaged in bags of $250$ $g$.
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Whatever is left will be packaged in bags of $100$ $g$.
If they have received $12$ sacks of one myriagram each, how many bags of each type will they obtain?
A two-pan balance 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, another $25$ $dag$…
A two-pan balance 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, another $25$ $dag$ weight, and the same honey jar but empty. Calculate the mass of the honey contained in the jar and the mass of the empty jar separately, expressing both results in grams.