Vaša košarica je prazna
Measurement for comparison
\latex{ 1 } jug of wine
\latex{ 1 } bag of wheat
\latex{ 1 } inch of string
When you measure something, you have to choose your unit of measurement. The best way to do this is to use one that is universally agreed upon so that comparisons are easy and practical for everyone.
In the Middle Ages, the development of agriculture, industry and trade brought about the need for commonly understood measurements of length, area, volume, mass and time.
Quantities are measured in appropriately chosen units.
volume
\latex{ 1 } \latex{ litre }
\latex{ 1 } \latex{ litre }
time
\latex{ 1 } \latex{ hour }
\latex{ 1 } \latex{ hour }
lenght
\latex{ 19 } \latex{ centimetres }
\latex{ 19 } \latex{ centimetres }
area
\latex{ 8 } \latex{ hectares }
\latex{ 8 } \latex{ hectares }
mass
\latex{ 300 } \latex{ grams }
\latex{ 300 } \latex{ grams }
When measuring, quantities are compared on the basis of a chosen unit. The comparison of quantities is precise when you provide both the value and the unit of your measurement.
quantity
length
\latex{ 115 }
\latex{ cm }
value
unit
Comparing amounts
- Which cake is the heaviest if the unit is the same but the values are different?
\latex{\textcolor{#0000ff}{100\,}g }
\latex{\textcolor{#0000ff}{115\,}g }
\latex{\textcolor{#0000ff}{120\,}g }
\latex{\textcolor{#0000ff}{125\,}g }
<
<
<
If the unit is the same, the larger the preceding number, the larger the quantity.
- Which child is the oldest if the value is the same but the units are different?
\latex{ 10 } days
\latex{ 10 } months
\latex{ 10 } years
<
<
If the value is the same, the larger the unit, the larger the quantity.
Example 1
You have two hundred €\latex{ 1 } coins in your pocket. Exchange them first for €\latex{ 20 } bills, and then for €\latex{ 100 } bills.
Solution
\latex{\textcolor{#0000ff}{200}} €\latex{ 1 } coins
\latex{\textcolor{#0000ff}{20}} €\latex{ 10 } bills
\latex{\textcolor{#0000ff}{2}} €\latex{ 100 } bills
=
=
\latex{ \textcolor{#ff0000}{\times 10}}
\latex{ \textcolor{#ff0000}{\times 10}}
\latex{ \textcolor{#ff0000}{\div 10}}
\latex{ \textcolor{#ff0000}{\div 10}}
You must multiply the unit and divide the value by the same number to maintain the same quantity. Similarly, a decrease in units must be accompanied by an increase in value.
Example 2*
There are three seemingly similar coins: an albe, a balge and a calbe. One of the coins is heavier than the other two (which weigh the same). You have a scale that shows how heavy the coins are compared to one another. Can you determine which coin is the heaviest of all from one single measurement?
Solution
Put an albe and a balge on either side of the scale.
There are three possible outcomes:
If albe > balge,
than albe is the heaviest
(balge = calbe).
than albe is the heaviest
(balge = calbe).
If albe < balge,
then balge is the heaviest
(albe = calbe).
then balge is the heaviest
(albe = calbe).
If albe = balge,
then calbe is left out,
so it is the heaviest coin.
then calbe is left out,
so it is the heaviest coin.
This means you can determine which coin is the heaviest of the three in a single measurement.
In the previous measurement, we compared values in relation to each other. Establishing a sequential order based on a given parameter is also a measurement. An example of this is competitions and championships.
Exercises
{{exercise_number}}. Get one of each of the coins used in your country. Arrange them in order of physical size. Is the order of size the same as the order of value?
{{exercise_number}}. Ant has €\latex{ 1 } and Mouse has €\latex{ 10 } coins. Rabbit has €\latex{ 100 } and Elephant has €\latex{ 1,000 } banknotes. How many coins or banknotes do they need to pay for the following amounts?
a) €\latex{ 42 }
b) €\latex{ 163 }
c) €\latex{ 546 }
b) €\latex{ 163 }
c) €\latex{ 546 }
a) €\latex{ 140 }
b) €\latex{ 340 }
c) €\latex{ 972 }
b) €\latex{ 340 }
c) €\latex{ 972 }
a) €\latex{ 65 }
b) €\latex{ 480 }
c) €\latex{ 1,000 }
b) €\latex{ 480 }
c) €\latex{ 1,000 }
a) €\latex{ 1,400 }
b) €\latex{ 16,000 }
c) €\latex{ 120,000 }
b) €\latex{ 16,000 }
c) €\latex{ 120,000 }
{{exercise_number}}. Will, Tom, Matt, David and John took part in a chess competition.
- Matt beat Will, Tom, David and John.
- John beat Will and Tom.
- David beat Tom, Will and John.
- Will beat Tom.
What is the final ranking in the competition?
{{exercise_number}}. Pete and Gabe weigh more together than Louie and Zoe. If Gabe and Zoe were to switch places, the scale would tip the other way. Who weighs more, Gabe or Zoe?
{{exercise_number}}. In a race, Anna crossed the finish line before Berta, but after Cecilia. Dan and Cecilia did not cross the finish line together. Anna and Dan did not cross the finish line just before or just after Greg. What could be the possible order in which the runners crossed the finish line? Find all the solutions.
{{exercise_number}}. Make a list of at least \latex{ 8 } items of food that your family bought last week. Write down (or at least estimate) how much each cost, then calculate how much money was paid at the shop.
{{exercise_number}}. Put the following in ascending order. If necessary, look up the information or make a guess based on what you already know.
- China, Luxembourg, Hungary, Germany – by the number of inhabitants
- Paris, Tokyo, London, New York – by the number of inhabitants
- Alexander the Great, Charlemagne, Napoleon, Churchill – by the year of their birth
- bicycle, airplane, car, scooter, bus – by their prices
{{exercise_number}}. For the following quantities, write the units in green and the values in blue.
- \latex{ 3 } \latex{ litres } of milk
- \latex{ 25\,dkg } of salami
- \latex{ 40\,dkg } of cheese
- half a \latex{ kilo } of bread
- \latex{ 1 } \latex{ litre } of detergent
- \latex{ 240 } \latex{ minutes } of videotape
- a \latex{ 50\,cm } wide and \latex{ 100\,cm } long towel
{{exercise_number}}. The top five downhill skiers at the Winter Olympics were Finland, Germany, Austria, Switzerland and France, in that order. After each of them had skied down, the scoreboard showed their position at the given moment. What was the order of the top five competitors if two \latex{ 1 }-s and three \latex{ 3 }-s appeared in succession?
{{exercise_number}}. Out of nine similar coins, one is lighter than the others. At least how many measurements do we have to take with a scale to find the lighter coin?
{{exercise_number}}. Collect all the different units (such as length or weight) that you have learnt in the lower grades. List all their units of measurement in ascending order.
Quiz
Which is worth more? \latex{ 2\,kg } of €\latex{ 1 } coins or \latex{ 1\,kg } of €\latex{ 2 } coins?
