Exercises: Length

In the study of mathematics and physics, it is essential to distinguish between what we measure and the unit we use. Classify the following concepts in your notebook as either a quantity or a unit of measurement:

• Length

• Metre ($m$)

• Centimetre ($cm$)

• Width of a table

• Kilometre ($km$)

• Distance between two cities

To measure accurately, we must choose the most suitable unit. State which metric unit ($mm$, $cm$, $m$ or $km$) you would use to measure the following objects:

1. a) The length of a paperclip.
2. b) The height of your best friend.
3. c) The distance a school bus travels between two towns.
4. d) The thickness of a latest-generation smartphone screen.

The metric system is a base-10 decimal system because each unit is $10$ times larger than the previous one. Complete the following equalities by converting to the indicated unit:

1. a) $7\text{ m} = \dots\dots\dots\text{ dm}$
2. b) $25\text{ cm} = \dots\dots\dots\text{ mm}$
3. c) $1,5\text{ m} = \dots\dots\dots\text{ cm}$
4. d) $0,8\text{ dm} = \dots\dots\dots\text{ mm}$

When measuring large distances, we use multiples of the metre. Perform the following conversions:

1. a) $3.000\text{ m} = \dots\dots\dots\text{ km}$
2. b) $450\text{ m} = \dots\dots\dots\text{ hm}$
3. c) $12\text{ dam} = \dots\dots\dots\text{ m}$
4. d) $0,5\text{ km} = \dots\dots\dots\text{ hm}$

Three friends compare the length of their headphone cables to see which one is longer. The measurements are:

• Marcos: $1,2\text{ m}$

• Lucía: $125\text{ cm}$

• Hugo: $11,5\text{ dm}$

Express all the measurements in centimeters and order them from smallest to largest using the symbol $<$.

A stage of a hiking route through the Sierra de Guadarrama is divided into two sections. The first section measures $3\text{ km y } 450\text{ m}$, and the second section measures $2.100\text{ m}$. How many kilometres is the complete stage?

Sara is designing a case for her tablet. She uses a ruler and observes that the length of the tablet is $24,5\text{ cm}$. If she wants to add a protective border of $5\text{ mm}$ to each of the two ends of the length, what will be the total length of the case in centimetres?

Calculate the perimeter (the sum of all its sides) of a five-a-side football pitch measuring $40\text{ m}$ in length and $200\text{ dm}$ in width. Express the final result in metres.

(Hint: Remember that to add measurements, they must all be in the same unit).

Mateo has to cut a strip of wood for a Technology project and needs exactly $4\text{ m and } 5\text{ cm}$. To write down the measurement in his notebook in a single unit (as a simple quantity in metres), he directly writes $4,5\text{ m}$.\na) Explain with reasons why Mateo's notation is incorrect, supporting your answer with the place value of each digit in the metric system.\nb) What is the correct expression in metres of the measurement he needs?\nc) What actual length in centimetres would the strip have if Mateo cut the $4,5\text{ m}$ he wrote down by mistake? Calculate the difference in centimetres between what he needs and what he would cut.

In a secondary school, they want to install protective skirting boards in one of the 1st year ESO classrooms. The classroom is rectangular and measures $82\text{ dm}$ in length and $580\text{ cm}$ in width. The entrance door is $9\text{ dm}$ wide and, obviously, will not have any skirting board.\na) Before calculating precisely, mentally estimate how many metres of skirting board will be needed by rounding the wall measurements to the nearest metre. Do you think buying a standard pack of $25\text{ m}$ of skirting board will be enough? Justify your answer.\nb) Express all dimensions in metres and calculate the exact length of skirting board needed.\nc) Compare your previous estimate with the exact result and explain whether the initial estimate was sufficient to make a purchasing decision.

Three friends are comparing the distances they ran this morning while training for a charity race. Lucía says she ran $4,2 \text{ km}$, Marcos claims to have covered $38 \text{ hm}$ and $15 \text{ dam}$, and Sofía recorded a total of $4.050 \text{ m}$ on her smartwatch.

Order the three friends from greatest to shortest distance covered, expressing all measurements in metres to compare them.

A delivery drone completes a journey divided into three stages. In the first stage, it covers a distance of $1 \text{ km}$ and $25 \text{ m}$. In the second, it flies $85 \text{ hm}$ and $7 \text{ dam}$. In the final stage, it covers another $450 \text{ m}$.

Calculate the total distance of the journey, expressing the result as a single unit in kilometres.

In a tech shop, monitor screens are measured by the length of their diagonal in inches. We know that $1 \text{ pulgada} = 2,54 \text{ cm}$.

If we buy a monitor whose diagonal measures $27 \text{ pulgadas}$, what is its approximate measurement in millimetres?

A carpenter has a wooden batten $3 \text{ m}$ long. He needs to cut pieces of $35 \text{ cm}$ each to make the legs for some stools.

1. a) How many full pieces of $35 \text{ cm}$ can he get from the batten?
2. b) How many millimetres of wood will he have left over after making the cuts?

The perimeter of a go-kart track measures $0,85 \text{ km}$. During a race, a driver completed $12$ laps of the track.

Calculate the total distance covered by the driver and express it in compound form (using kilometres, hectometres and decametres).

A biologist is studying the growth of a root under a microscope. On the first day, the root measured $12 \text{ mm}$. After one week, the root measures $4,5 \text{ cm}$.

How many micrometres ($\mu m$) has the root grown in total during that week? (Remember that $1 \text{ mm} = 1.000 \mu m$).

To install optical fibre in a building, $3$ rolls of cable of $1,2 \text{ hm}$ each are needed. If the technician has already used $245 \text{ m}$ of cable, how many decametres of cable does he still have available?

An ant walks along the edge of a rectangular table measuring $1,2 \text{ m}$ long and $80 \text{ cm}$ wide. If the ant completes $5$ full laps around the edge of the table:

1. a) How many metres does it cover in total?
2. b) Express the previous result in hectometres.

In the long jump event during the school's sports week, Hugo recorded a mark of $3\text{ m and } 8\text{ cm}$, while his classmate Daniela achieved a jump of $2\text{ m, } 7\text{ dm and } 5\text{ cm}$. To find out by how much Hugo beat Daniela's mark, the teacher suggests three possible ways:\n- Method A: Convert both marks to metres using decimal numbers and subtract.\n- Method B: Express both measurements in centimetres as whole numbers and subtract.\n- Method C: Work directly in compound units by subtracting units of the same order ($3\text{ m} - 2\text{ m}$, etc.) and converting units when necessary.\na) Solve the subtraction using Method A and Method B, checking that you obtain the same result.\nb) Analyse the advantages and disadvantages of the three methods and justify which one you consider most advisable to avoid calculation errors in this specific case.

The official road book of a cycling stage running along mountain roads indicates a total planned distance of $164,8\text{ km}$. The route is divided into four sectors:\n- Sector 1 (start and flat): $42\text{ km and } 600\text{ m}$.\n- Sector 2 (mountain pass): $185\text{ hm}$.\n- Sector 3 (valley): $6.400\text{ dam}$.\n- Sector 4 (final climb to the finish line): length pending confirmation.\nDue to unforeseen roadworks in the valley, the organisers have to add a mandatory detour of $3.250\text{ m}$ within Sector 3.\na) Set up the necessary operations in the same unit to determine the original length of Sector 4 before the roadworks, and write the result in compound units using kilometres and metres.\nb) Model the final length the complete stage will have after incorporating the detour, expressing the final result in hectometres ($hm$).

A hiking club is designing a new circular route in the Pyrenees. On the map (scale $1:25.000$), the route is divided into three sections: the first measures $12,4$ $cm$, the second $85$ $mm$ and the third $0,06$ $dm$. If a hiker walks at a constant speed and each of their steps measures exactly $75$ $cm$, calculate how many steps they will take to complete the entire route in real life. Also express the total distance of the route in hectometres ($hm$).

In a nanotechnology laboratory, a scientist observes a synthetic fibre whose thickness is $450$ nanometres ($nm$). For an experiment, they need to place these fibres side by side to cover a distance of $0,27$ millimetres ($mm$). Knowing that $1$ micrometre ($\mu m$) is $1.000$ $nm$ and that $1$ $mm$ is $1.000$ $\mu m$, how many fibres will the scientist need to place? If each fibre costs $0,002$ euros, what would be the total cost of the sample?

In the past in Spain, the "Castilian vara" was used, which is approximately equal to $83,59$ $cm$. A farmer reads in an old document that his land has a frontage of $120$ varas. To comply with current regulations, he must install a fence costing $15,50$ euros per linear metre. However, for every $5$ decametres ($dam$) of fence, the shop gives him a discount of $10\%$ on the price of those metres. How much money will it cost him to fence the entire frontage of his land?

A racing drone completes a circuit shaped like an irregular hexagon. The lengths of its sides are: $0,15$ $km$, $2,4$ $hm$, $35$ $dam$, $1.200$ $dm$, $8.500$ $cm$ and $95.000$ $mm$. The drone must complete a total of $15$ laps of the circuit. What is the total distance the drone will travel in kilometres? If the drone's battery lasts long enough to cover $185$ hectometres, will it be able to finish all the laps or will it run out of battery first? Justify your answer with calculations.

A Maths textbook has a total thickness (excluding the covers) of $2,4$ $cm$. If the book has exactly $400$ pages ($200$ sheets), what is the thickness of a single sheet expressed in microns ($\mu m$)? Knowing that a human hair has an average thickness of $0,08$ $mm$, how many human hairs would need to be stacked to equal the thickness of a single sheet of this book?

A school merchandise designer needs to edge $12$ identical circular banners with decorative ribbon. The perimeter of each banner is $37,5$ decimetres ($dm$). The ribbon is sold in rolls of $0,5$ hectometres ($hm$).

1. a) What is the minimum number of full rolls the designer must buy so that she does not run out of ribbon?
2. b) Once the $12$ banners are finished, how many centimetres of ribbon will be left over in total?
3. c) If each metre of ribbon costs $1,20$ euros, what is the price of a full roll?

The distance between two cities, A and B, is $145,6$ $km$. Exactly at the midpoint of the road connecting them, there is a signpost indicating the distance to a third city C, located $34.500$ metres from that midpoint. If a car travels from A towards B and stops after covering $820$ hectometres ($hm$):

1. a) How far is it from city B?
2. b) How far is it from the signpost indicating the way to city C?

Express all results in metres.

An artificial satellite orbits the Earth at a constant altitude of $350$ $km$. However, due to atmospheric drag, it loses $450$ metres of altitude each day.

1. a) At what altitude in kilometres will the satellite be after $3$ weeks in orbit?
2. b) If the satellite stops working when it falls below $330$ $km$ in altitude, for how many full days will it remain operational?
3. c) Express the satellite's initial altitude as a compound expression using kilometres, hectometres and metres.

A communal plot is quadrilateral in shape and its total perimeter measures exactly $0,15\text{ km}$. We do not know all the direct measurements, but the following data are available:\n- Side $A$ measures $35\text{ m and } 40\text{ dm}$.\n- All four sides measure an exact integer number of metres.\n- Side $D$ has a length that, expressed in decametres ($dam$), is an exact integer, and it is known to measure more than $20\text{ m}$ but less than $45\text{ m}$.\n- Side $B$ measures exactly twice side $C$.\na) Deduce with reasons the only possible length for side $D$, ruling out the options that do not satisfy all the conditions.\nb) From the value found for $D$, calculate the exact measure of sides $B$ and $C$, and express them in compound units combining decametres and metres.\nc) Check that the sum of the four sides reproduces exactly the initial perimeter in kilometres.

A team from the National Geographic Institute is marking a mountain trail with a total length of $14,4\text{ km}$ between two mountain huts. For safety and visibility in foggy conditions, stone cairns are installed along the path following these rules:\n- Lower section (the first $4\text{ km and } 8\text{ hm}$): the path is wide and a cairn is placed every $120\text{ m}$.\n- Middle section (from the end of the lower section to kilometre marker $11,2\text{ km}$ of the trail): a cairn is placed every $8\text{ dam}$.\n- High summit section (from kilometre marker $11,2\text{ km}$ to the end of the trail): the path is stony and a cairn is placed every $2.500\text{ cm}$.\nA cairn is placed at the start of the trail ($0\text{ m}$), another at each section change, and another at the final hut.\na) Calculate the exact length of each of the three sections expressed in metres.\nb) Determine the total number of cairns that will be needed to mark the entire trail from start to finish.\nc) One of the hikers states: «Since the high summit section is the shortest of the three in distance, it is surely the section where the fewest cairns need to be carried and placed». Critically and numerically analyse whether their statement is true or false, justifying which quantity has a greater influence on the result.