Exercises: Mass

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.

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:

1. a) $45\text{ kg} = \dots\dots\dots\text{ g}$
2. b) $1.200\text{ mg} = \dots\dots\dots\text{ g}$
3. c) $0,5\text{ hg} = \dots\dots\dots\text{ dag}$
4. d) $34,7\text{ dg} = \dots\dots\dots\text{ cg}$

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:

1. A worker ant.

2. A full can of soda.

3. A laptop.

4. A delivery truck.

Possible measurements: $2\text{ kg}$, $8\text{ t}$, $3\text{ mg}$, $330\text{ g}$.

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: $4.200\text{ dg}$

• 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.

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?

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 $200\text{ g}$.

• 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:

1. a) $3\text{ kg y } 450\text{ g}$ $\rightarrow$ in grams.
2. b) $5\text{ dag, } 2\text{ g y } 8\text{ dg}$ $\rightarrow$ in decigrams.
3. c) $1\text{ t y } 200\text{ kg}$ $\rightarrow$ in kilograms.
4. d) $4\text{ hg y } 5\text{ cg}$ $\rightarrow$ in centigrams.

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 taking to class today:

• Books and notebooks: $2,8\ kg$

• Pencil case with materials: $350\ g$

• English dictionary: $6,5\ hg$

• 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 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: $1.250\ dg$

• 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).

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.

1. a) How many sacks are needed to transport the entire harvest?
2. 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?

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?

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.

In a bulk candy shop, the price of marshmallows is $1,20\ \$$ per hectogram ($hg$).

1. a) How much will a bag containing $250\ g$ of marshmallows cost?
2. b) If we have paid $6\ \$$ for another bag of candy, how many grams of candy have we bought?

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 ($mg$):

• Camera sensor: $0,0042\ kg$

• Main microchip: $1,85\ g$

• Fixing screw: $85\ cg$

• Charging connector: $12,4\ dg$

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 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.

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.

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.

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?

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 medals he will be able to make?

• 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?

• How much silver will he have left over in milligrams?

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:

• Half of the rice will be packaged in bags of $500$ $g$.

• Two-fifths of the remaining rice will be packaged in bags of $250$ $g$.

• 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?

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 centimeter ($cm^3$)?

• 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 $V = arista \cdot arista \cdot arista$)

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.