Lesson: Whole Numbers and Word Problems Using the Unitary Method (Grade 5)
Hello everyone! Have you ever wondered: if \(3\) notebooks cost \(36\) baht, how much would you have to pay for \(5\) of the same notebooks? Or if \(4\) kilograms of oranges cost \(120\) baht, how much would \(10\) kilograms cost? For everyday math problems like these, finding the answer is super easy with a magical mathematical tool called the "Unitary Method" (Rule of Three). If it feels a bit tricky at first, don't worry—let's take it step by step together!
Note: To build upon this concept for fractions, decimals, or percentages, you will be able to explore more in the upcoming Grade 5 lessons.
---1. What Is the Unitary Method?
The traditional Thai term "Banyat Traiyang" sounds quite formal, but it comes from the word "trai", which means three (also known in English as the Rule of Three or the Unitary Method).
The Unitary Method is a way to find an unknown quantity from the relationship between two quantities. The problem gives us \(3\) known values to help us find the \(4\text{th}\) unknown value that we want to solve for.
Did you know?
In everyday life, we use the unitary method all the time without even realizing it! For example, calculating prices at the grocery market, figuring out fuel consumption based on travel distance, or adjusting recipe ingredients to bake the exact number of cookies we want.
2. The Key Secret: Always Find the Value of "1 Unit" First
The heart of solving problems with the unitary method at this level is "finding the value of a single unit (1 unit)" through two simple steps:
1. Step 1 (Find 1 unit using division): Take the total value given in the problem and divide it to find the value of 1 single unit.
2. Step 2 (Find the desired quantity using multiplication): Multiply the value of 1 unit by the number of units the question asks for.
A quick visual example:
\(2\) boxes contain a total of \(10\) snacks.
- \(1\) box contains \(10 \div 2 = 5\) snacks (Division step)
- \(4\) boxes will contain \(4 \times 5 = 20\) snacks (Multiplication step)
3. The 3-Line Working Method
To show your working neatly and correctly in unitary method problems, we write down 3 lines using one very important golden rule:
"Keep what the problem gives on the LEFT, and put what the problem asks for on the RIGHT."
The Magic 3-Line Structure:
Line 1: Write down the full relationship given by the problem (Initial quantity on the left — Resulting quantity on the right).
Line 2: Find the quantity for 1 unit (written as a division or in fraction form).
Line 3: Find the quantity the problem asks for (multiply the target amount by the value of 1 unit).
4. Let's Work Through Some Examples!
Example 1: Buying and Selling Items
Problem: \(4\) pencils cost \(24\) baht. If you buy \(7\) of the same pencils, how much do you have to pay?
Problem Analysis:
- The problem asks for the "amount of money (baht)", so we put baht on the right.
- The given quantity being compared is the "number of pencils", so we put pencils on the left.
Solution:
Line 1: \(4\) pencils cost \(24\) baht
Line 2: \(1\) pencil costs \(24 \div 4 = 6\) baht (or written as a fraction: \(\frac{24}{4} = 6\) baht)
Line 3: \(7\) pencils cost \(7 \times 6 = 42\) baht
Answer: ๔๒ baht (or 42 baht)
Example 2: Work Rates and Time
Problem: Somchai went jogging for \(3\) days and covered a total distance of \(15\) kilometers. If he continues running at the same rate for \(8\) days, how many kilometers will he run in total?
Problem Analysis:
- The problem asks for "distance (kilometers)", so put kilometers on the right.
- The quantity used for comparison is "number of days (days)", so put days on the left.
Solution:
Line 1: In \(3\) days, he runs a distance of \(15\) kilometers
Line 2: In \(1\) day, he runs a distance of \(15 \div 3 = 5\) kilometers
Line 3: In \(8\) days, he runs a distance of \(8 \times 5 = 40\) kilometers
Answer: ๔๐ kilometers (or 40 kilometers)
Example 3: Swapping Sides (Finding the Quantity from the Cost)
Sometimes, a question asks the reverse! Make sure to pay close attention to which unit goes on the left and which goes on the right.
Problem: \(5\) cartons of fresh milk cost \(60\) baht. If you have \(180\) baht, how many cartons of milk can you buy?
Problem Analysis:
- The problem asks for the "number of cartons (cartons)", so this time we place cartons on the right.
- The quantity given for comparison is the "amount of money (baht)", so place baht on the left.
Solution:
Line 1: With \(60\) baht, you can buy \(5\) cartons of milk
Line 2: With \(1\) baht, you can buy \(\frac{5}{60}\) cartons of milk
Line 3: With \(180\) baht, you can buy \(180 \times \frac{5}{60} = \frac{180 \times 5}{60} = 15\) cartons of milk
Answer: ๑๕ cartons (or 15 cartons)
Common Mistakes to Watch Out For!
1. Swapping the Left and Right sides: Always remember the golden rule: "The unit you want to find must always be on the right-hand side." If the problem asks for baht, put baht on the right. If it asks for items, put items on the right.
2. Mixing up the dividend and divisor in Line 2: When finding the value of 1 unit, the number on the right is always the dividend (numerator), and the number on the left is always the divisor (denominator). For example, if the right is \(24\) and the left is \(4\), it must be \(24 \div 4\), never \(4 \div 24\).
3. Forgetting to write the units: Don't forget to clearly label the units on every line! This helps you check whether you placed the quantities on the correct sides.
Key Takeaways
- The Unitary Method is a tool for solving word problems where \(3\) values are given to find a \(4\text{th}\) unknown value.
- Memorize the 3-line steps:
Line 1: State what the problem tells you (Target unit goes on the right).
Line 2: Find the value of 1 unit (Divide the right number by the left number).
Line 3: Find the target quantity (Multiply the new quantity by the value of 1 unit).
- Always check if your answer is reasonable! For example, if you buy more items, the total price should increase too!