【Grade 5 Math】Quantity per Unit: Solving "Which one is more crowded?"
Hello! Today, let's study a very important topic in 5th-grade math: "Quantity per Unit."
We will learn how to use numbers to clearly decide which is "the better deal" or "the more impressive one" in everyday situations, such as "Which classroom is more packed?" or "Which supermarket has the cheaper meat?"
It might feel a little tricky at first, but once you grasp the knack, it becomes as fun as solving a puzzle. Let's take it step by step!
1. Let's compare "crowdedness"
For example, how should you compare the "crowdedness" of two rooms?
- Pattern A: If the rooms are the same size, the one with more people is more crowded.
- Pattern B: If the number of people is the same, the smaller room is more crowded.
But it's difficult when "both the size and the number of people are different." That is where "Quantity per Unit" comes in!
★ Two ways to compare
The key is to set one of the values to "1".
① How many people are there per \(1 m^2\)? (People \(\div\) Area)
The larger the result of the calculation, the more crowded it is.
② How many \(m^2\) of space is there per person? (Area \(\div\) People)
The smaller the result of the calculation, the more crowded it is (because it means each person has less space).
【Point!】
When you want to find the value "per \(1\)", divide by the number that represents the unit you want to make "1"!
(Example: If you want to know the value per \(1 m^2\), divide by the area.)
2. The Formula for Quantity per Unit
This way of thinking can be used in many situations beyond just crowdedness. Let's learn the basic form.
\( \text{Quantity per unit} = \text{Total amount} \div \text{How many units} \)
(Example 1) A pump outputs 120L of water in 3 minutes. How many liters come out per minute?
\( 120 \div 3 = 40 \)
Answer: \(40L\)
(Example 2) If 4 apples cost 500 yen, how much is it per apple?
\( 500 \div 4 = 125 \)
Answer: \(125\) yen
【Common Mistake】
If you are confused about which number to divide by, ask yourself, "What do I want to make '1'?" If you want to know the price per apple, divide by the number of apples (4). Conversely, if you want to know how many apples you can buy for 1 yen, divide by the price (500).
3. Population Density
"Population density," which you also learn about in Social Studies, is a type of quantity per unit.
\( \text{Population density} = \text{Population} \div \text{Area} (km^2) \)
This represents "how many people live per \(1 km^2\)." The higher the number, the busier and more densely populated the area is.
【Trivia】
The population density of Japan is about 330 people, but there are countries in the world with much higher densities, and others—large countries—with less than one person per square kilometer. By comparing by unit, we can compare accurately even if the countries have different sizes!
4. Average
"Average" is also a kind of quantity per unit. It is the size of something when you smooth out (level out) uneven values so that they are all the same size.
\( \text{Average} = \text{Total} \div \text{Number of items} \)
★ Steps to find an average
- First, add everything together to get the "Total."
- Divide that total by the "number" of things you added.
(Example) Your test scores were 80, 90, and 70. What is the average score?
Step 1: \( 80 + 90 + 70 = 240 \) (Total)
Step 2: \( 240 \div 3 = 80 \)
Answer: Average \(80\) points
【Attention!】
If there is a time when the record is "0," you must still include that in the number of items (the divisor). Don't pretend it didn't happen!
5. Conclusion: The Trick to Understanding!
When solving quantity per unit problems, try chanting this mantra in your head:
"If you want to know the amount per '1', divide by that unit!"
- Want to know per \(1m\)? \(\rightarrow\) Divide by the meters.
- Want to know per \(1\) hour? \(\rightarrow\) Divide by the hours.
- Want to know per \(1kg\)? \(\rightarrow\) Divide by the kilograms.
If you follow this, you will never be confused about what to divide by. The calculations might feel tough at first, but once you can set up the equation, all that's left is to calculate carefully. Keep going one step at a time! I'm cheering for you!
★ Today's Summary (Key Takeaway)
・Quantity per unit is a handy tool for comparing things that are different.
・To find the "amount per 1," divide by that number.
・Average = Total \(\div\) Number of items.