Lesson: Ratios, Scales, and Percentages (Grade 6 / Prathom 6)
Hello everyone in Grade 6! In this lesson, we will explore ratios, scales, and percentages. These concepts are fun and super relevant to our everyday lives, such as mixing juice drinks, reading maps, or calculating sale discounts. If it feels a bit tricky at first, don't worry—let's take it step by step together!
1. Getting to Know "Ratios"
A ratio is a relationship that compares two or more quantities, which can have either the same unit or different units.
Writing a Ratio
We use the colon symbol \(:\) to represent the word "to".
When comparing quantity \(a\) to quantity \(b\), we write it as \(a : b\) or as a fraction \(\frac{a}{b}\) (read as "\(a\) to \(b\)").
Real-Life Example:
If Mom makes strawberry syrup drink using \(1\) cup of concentrated syrup to \(4\) cups of water,
we write the ratio of the amount of concentrated syrup to the amount of water as \(1 : 4\).
Important Note About Units:
- Same units: You do not need to write the units; for example, the ratio of pencils to erasers is \(3 : 2\).
- Different units: You must always include the units; for example, the ratio of eggs in units to price in Baht is \(10\text{ eggs} : 45\text{ Baht}\).
Common Mistake: Swapping the order of numbers! The order of quantities asked in the problem is very important. For example, a ratio of teachers to students of \(1 : 25\) is not the same as \(25 : 1\), because the meaning changes completely! Always check which quantity is mentioned first.
2. Equivalent Ratios
Equivalent ratios are ratios that express the same proportional relationship. You can find them by multiplying or dividing both terms (the first and the second numbers) by the same non-zero whole number.
Method 1: Multiplying by the Same Number (Scaling Up)
When we multiply both the first and the second quantities by the same non-zero whole number, the resulting new ratio is equal to the original ratio.
Example: Starting with the ratio \(2 : 3\)
Multiply both numbers by \(2\): \((2 \times 2) : (3 \times 2) = 4 : 6\)
Multiply both numbers by \(5\): \((2 \times 5) : (3 \times 5) = 10 : 15\)
Therefore, \(2 : 3 = 4 : 6 = 10 : 15\)
Method 2: Dividing by the Same Number (Simplifying / Scaling Down)
When we divide both the first and the second quantities by a common factor (a non-zero whole number that divides evenly into both), the new ratio is equal to the original ratio.
Example: Starting with the ratio \(12 : 18\)
Divide both numbers by \(6\): \((12 \div 6) : (18 \div 6) = 2 : 3\)
Therefore, \(12 : 18 = 2 : 3\)
Checking for Equivalent Ratios: We can check if two ratios are equivalent by simplifying both to their simplest form, or by using cross-multiplication: \(a : b = c : d\) is true when \(a \times d = b \times c\).
3. Scale
A scale is a ratio that shows the comparison between the distance on a drawing or map and the actual distance.
Core Formula: Scale = Distance on drawing : Actual distance
Example: A floor plan specifies a scale of \(1\text{ cm} : 5\text{ m}\).
This means that measuring \(1\text{ centimetre}\) on the floor plan represents \(5\text{ metres}\) in real life.
Did you know? When a scale is written purely as numbers without units, such as \(1 : 1,000\), it means both parts share the exact same unit. For instance, \(1\text{ centimetre}\) on the map represents \(1,000\text{ centimetres}\) (or \(10\text{ metres}\)) in real life!
Example of Solving a Scale Problem:
Problem: A map shows the distance from home to school as \(4\text{ centimetres}\) long using a scale of \(1\text{ cm} : 2\text{ km}\). What is the actual distance?
Solution:
A distance of \(1\text{ centimetre}\) on the map represents an actual distance of \(2\text{ kilometres}\).
A distance of \(4\text{ centimetres}\) on the map represents \(4 \times 2 = 8\text{ kilometres}\).
Answer: The actual distance from home to school is \(8\text{ kilometres}\).
4. Solving Ratio Word Problems
To solve word problems involving ratios, we can use our knowledge of equivalent ratios or apply the unitary method.
Word Problem Example:
Problem: A shop arranges gift sets using a ratio of notebooks to pens of \(3 : 5\). If the shop uses \(24\) notebooks in total, how many pens are needed?
Step-by-step Solution:
1. Write down the given ratio: \(\text{Notebooks} : \text{Pens} = 3 : 5\)
2. Identify what is given: There are \(24\) notebooks.
3. Look at the number of notebooks: From \(3\) notebooks to \(24\) notebooks, the value is multiplied by \(8\) (since \(24 \div 3 = 8\)).
4. Multiply the number of pens by that same value: \(5 \times 8 = 40\).
Answer: A total of \(40\) pens are needed.
5. Percentages and Solving Percentage Problems
A percentage (written with the symbol \(\%\)) is a ratio that compares any number to \(100\).
For example, \(25\%\) has the exact same meaning as the ratio \(25 : 100\), which can also be written as the fraction \(\frac{25}{100}\).
Calculating a Percentage of a Whole Number:
Remember this simple rule: in math word problems, the word "of" usually means multiplication!
Example: Find \(20\%\) of \(150\).
Working out: \(20\% = \frac{20}{100}\)
So we calculate: \(\frac{20}{100} \times 150 = \frac{20 \times 150}{100} = 30\)
Answer: \(30\)
Discount Word Problems:
Problem: A shirt is marked at \(400\) Baht. The shop offers a \(15\%\) discount. How much is the discount in Baht, and how much does the customer have to pay?
Solution:
1. Find the discount amount: \(15\%\) off the marked price of \(400\) Baht.
\(\text{Discount} = \frac{15}{100} \times 400 = 15 \times 4 = 60\text{ Baht}\)
2. Find the actual sale price: \(\text{Marked price} - \text{Discount} = 400 - 60 = 340\text{ Baht}\)
Answer: The shop offers a discount of \(60\) Baht, and the customer pays \(340\) Baht.
Profit and Loss Word Problems:
- \(10\%\) Profit: A cost price of \(100\) Baht earns a profit of \(10\) Baht, making the selling price \(100 + 10 = 110\) Baht.
- \(10\%\) Loss: A cost price of \(100\) Baht has a loss of \(10\) Baht, making the selling price \(100 - 10 = 90\) Baht.
Example: A merchant buys a watch at a cost price of \(800\) Baht and sells it for a \(20\%\) profit. How much money does the merchant sell the watch for?
Solution:
\(\text{Profit} = \frac{20}{100} \times 800 = 160\text{ Baht}\)
\(\text{Selling price} = \text{Cost price} + \text{Profit} = 800 + 160 = 960\text{ Baht}\)
Answer: The merchant sells the watch for \(960\) Baht.
Common Mistake: Not reading the question carefully! Some problems ask for the "discount amount," while others ask for the "final price to pay." Similarly, some ask for the "profit," while others ask for the "selling price." Always make sure you double-check what is being asked before writing down your final answer!
Quick Review
- Ratio (\(a : b\)): Compares two quantities. The order of numbers matters greatly!
- Equivalent Ratios: Found by multiplying or dividing both terms by the same non-zero number.
- Scale: The ratio comparing drawing distance to actual distance (Drawing distance : Actual distance).
- Percentage (\(\%\)): A comparison to \(100\), where \(a\%\) represents \(\frac{a}{100}\).
- When tackling word problems, take your time to analyse what is given and what is being asked. Plan your steps, calculate carefully, and practice often—maths will become easy and fun for everyone!