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 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 seems most realistic:

1. A worker ant.

2. A full can of drink.

3. A laptop.

4. A delivery lorry.

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

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:

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) $34.7\text{ dg} = \dots\dots\dots\text{ cg}$

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.

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.

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: $4,200\text{ dg}$

• Student C: $46\text{ dag}$

Express all measurements in grams and order the students from lowest to highest according to the mass they obtained.

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 $200\text{ g}$.

• A packet of sugar of $4.5\text{ hg}$.

What is the total mass of the ingredients in grams?

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 $1\text{ t} = 1,000\text{ kg}$).

Express the following quantities in simple form (using a single unit), converting them to the unit indicated in each case:

1. a) $3\text{ kg y } 450\text{ g}$ $\rightarrow$ into grams.
2. b) $5\text{ dag, } 2\text{ g y } 8\text{ dg}$ $\rightarrow$ into decigrams.
3. c) $1\text{ t y } 200\text{ kg}$ $\rightarrow$ into kilograms.
4. 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 ($mg$):

• Camera sensor: $0.0042\ kg$

• Main microchip: $1.85\ g$

• Fastening screw: $85\ cg$

• Charging connector: $12.4\ dg$

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?

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:

• Books and notebooks: $2.8\ kg$

• Pencil case with stationery: $350\ g$

• English dictionary: $6.5\ hg$

• 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.

In a candy shop that sells sweets by weight, the price of marshmallows is $\text{RD\$}\ 1.20$ per hectogram ($hg$).

1. a) How much will a bag containing $250\ g$ of marshmallows cost?
2. b) If we paid $\text{RD\$}\ 6$ for another bag of marshmallows, how many grams of marshmallows did we buy?

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: $1,250\ dg$

• 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).

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.

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

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 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.

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 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?

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 ($cm^3$)?

• 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 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.

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.

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 they can make?

• If one gram of aluminum costs $210$ pesos, how much will the material for all the manufactured pieces cost?

• How much aluminum will be left over in milligrams?

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:

• Half of the rice will be packed into $500$ $g$ bags.

• Two fifths of the remaining rice will be packed into $250$ $g$ bags.

• 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?