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 quantity, a unit of measurement or a measuring instrument:

Mass, Gram, Balance, Kilogram, Scales, Capacity, Tonne, Dynamometer.

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) $34,7\text{ dg} = \dots\dots\dots\text{ cg}$

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

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.

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?

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?

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.

A van has a maximum load capacity of $1,5$ metric tonnes ($t$). 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 $1\text{ t} = 1.000\text{ kg}$).

Sergio wants to prepare bread dough and needs to convert $4,5\ kg$ of flour to grams. In his notebook, he writes the following resolution: "To go from kilograms to grams I go down 3 steps on the metric system ladder, so I multiply by $30$: $4,5 \cdot 30 = 135\ g$".

1. a) Explain in your own words what conceptual error Sergio made regarding how the metric decimal system works.
2. b) Justify why he must multiply by $1\,000$ (that is, $10 \cdot 10 \cdot 10$) and not by $30$.
3. c) Correct Sergio's calculation and indicate the actual mass of flour, in grams, that he must use for the recipe.

In the eurozone, a $1$ euro coin has an official mass of $7,5\ g$. Imagine you have a bag containing exactly $60$ $1$-euro coins.

1. a) Before doing any exact calculation, round the mass of one coin to a simple whole number and mentally estimate approximately how many grams and how many kilograms the $60$ coins will weigh. Write down how you made that mental estimation.
2. b) Perform the exact calculation of the total mass of the coins and express the result in both grams ($g$) and kilograms ($kg$).
3. c) Compare your initial estimation with the exact result: was it a reasonable approximation? Explain whether your estimation was above or below the real value and why.

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.

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, $4,2$ tonnes 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?

Knowing that one cubic decimetre ($1\ dm^3$) of distilled water has a mass of exactly $1\ kg$, solve the following problem:

An aquarium is shaped like a cuboid 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?

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.

In a candy shop that sells sweets by weight, 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 paid $6\ €$ for another bag of marshmallows, how many grams of marshmallows did we buy?

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: $12,4\ dg$

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.

In the Natural Sciences laboratory, you must place three mineral samples on a tray that supports a maximum load of $1,5\ kg$. The masses of the samples are: pyrite ($520\ g$), quartz ($4,8\ hg$), and calcite ($55\,000\ cg$). To find out if the tray will support all three samples at the same time, two classmates propose different approaches:

• Rocío proposes converting all masses to grams ($g$) to add only whole numbers and compare the total with the capacity of the tray also expressed in grams.

• Irene proposes converting all masses directly to kilograms ($kg$) using decimal numbers and comparing the sum directly with $1,5\ kg$.

1. a) Solve the problem applying Rocío's method.
2. b) Solve the problem applying Irene's method.
3. c) Analyze both procedures: which of the two seems more practical to you and with less risk of error in this specific case? Justify your answer by evaluating working with whole numbers versus using decimal commas.

According to data on selective waste collection in Spain, it is estimated that a student generates an average of $35\ g$ of plastic waste and light packaging during a school day. In a secondary school, there are $500$ students.

1. a) Model the situation and calculate how many kilograms of plastic the students of the school generate in total in one school day and in a full school week (from Monday to Friday).
2. b) To store these containers, the school has a container whose maximum recommended load is $0,7\ q$ (metric quintals). Will it have enough capacity to accumulate the waste generated during two full school weeks (10 days) without being emptied? Justify your answer by indicating how many kilograms are left over or missing to reach the container's limit.
3. c) If after an environmental campaign the school reduces total daily generation to $14\ kg$, calculate how many grams of plastic each student saves per day on average compared to the initial situation.

A delivery lorry has a maximum load capacity of $3,5$ metric tonnes. 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.

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.

In an oil factory, a bottling machine fills $0,75$-litre bottles. We know that $1$ litre 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 tonnes? 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.

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?

The robotics club has an aluminium 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 aluminium costs $3,50$ €, how much will the material for all the manufactured pieces cost?

• How much aluminium 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?

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 centimetre ($cm^3$)?

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

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.

A metalsmith designs a ceremonial ingot made of three precious metals (gold, silver, and copper) whose combined total mass must be exactly $0,48\ kg$. Due to design requirements, the masses of the metals must satisfy the following conditions:

• The mass of silver must be four times the mass of copper.

• The mass of gold must exceed the mass of silver by exactly $60\ g$.

1. a) Express all quantities in the same unit and set up a step-by-step reasoning (or an equation) to deduce the exact mass, in grams, that each of the three metals must have in the ingot.
2. b) Check that the calculated masses satisfy all the imposed conditions and that their sum equals the total mass of the ingot in kilograms.
3. c) To calibrate the obtained portion of copper on a classic two-pan balance, the metalsmith only has weights of $1\ dag$, $1\ g$, $1\ dg$, and $1\ cg$. Determine the combination that uses the smallest possible number of weights to balance the calculated mass of copper, rounded to the nearest centigram.

An oil mill in Jaén prepares a shipment of olive oil in a truck whose tare weight (mass of the empty truck) is $3,8\ t$ and whose Gross Vehicle Weight Rating (GVWR) is $10\ t$. $16$ wooden pallets of $20\ kg$ each are loaded. On each pallet, $30$ cardboard boxes are placed, and each box contains $12$ bottles. Each full bottle of oil has a mass of $950\ g$ and each empty cardboard box weighs $25\ dag$.

The logistics assistant gives the following note to the carrier:

"Mass of bottles: $16 \cdot 30 \cdot 12 \cdot 950 = 5\,472\,000\ g = 54,72\ kg$.

Mass of boxes: $16 \cdot 30 \cdot 25 = 12\,000\ dag = 12\ kg$.

Mass of pallets: $16 \cdot 20 = 320\ kg$.

Total mass of cargo: $54,72 + 12 + 320 = 386,72\ kg = 0,38672\ t$.

Since $3,8\ t + 0,38672\ t \approx 4,19\ t$, the truck is almost empty and we can easily load twice as much".

1. a) Detect and explain the two serious unit conversion errors that the assistant made. Reason why their final result makes no physical sense at all for a load of thousands of bottles.
2. b) Carry out the correct calculation of the actual total mass of the loaded truck (tare weight, pallets, boxes, and bottles), expressing the result in both kilograms ($kg$) and metric tons ($t$).
3. c) Determine whether the truck can travel legally or if it exceeds the GVWR. If it does not exceed it, calculate accurately how many more full boxes could be loaded at most without exceeding the legal limit of $10\ t$.