Direct & Inverse Proportions
Direct Proportion
What is Direct Proportion?
Two quantities are in direct proportion if they increase or decrease together in the same ratio. When one quantity doubles, the other doubles; when one is halved, the other is halved.
If \(x\) and \(y\) are directly proportional, then \(\frac{x}{y} = k\) (constant), or \(x = ky\).
What is the Unitary Method?
The unitary method is a technique for solving proportion problems by first finding the value of one unit, then multiplying to find the required value.
Steps for Unitary Method:
- Find the value of one unit (divide the given quantity by the given number)
- Multiply by the required number of units
Real-Life Examples of Direct Proportion:
- Cost of items: more items → more cost (at same price)
- Distance and time (at constant speed): more time → more distance
- Work done and number of workers (at same rate): more workers → more work
Proportion Notation:
If \(a : b = c : d\), then \(a\), \(b\), \(c\), \(d\) are in proportion. We write \(a : b :: c : d\) and say "\(a\) is to \(b\) as \(c\) is to \(d\)".
In a proportion, product of extremes = product of means:
a \times d = b \times c
\]
If 8 books cost ₹240, find the cost of 5 books using the unitary method.
- Cost of 8 books = ₹240
- Cost of 1 book = \(240 \div 8 = ₹30\)
- Cost of 5 books = \(30 \times 5 = ₹150\)
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Answer: ₹150 **Example 2:** Check whether 3, 5, 12, 20 are in proportion. *Solution:* - Product of extremes = $3 \times 20 = 60$ - Product of means = $5 \times 12 = 60$ - $60 = 60$, so they are in proportion - **Answer:** Yes, they are in proportion **Example 3:** If 15 workers can build a wall in 20 days, how many workers are needed to build the same wall in 12 days? (Hint: This is inverse proportion, but check!) *Solution:* - More workers → fewer days (Inverse proportion) - Workers × Days = constant - $15 \times 20 = 300$ (total work) - Required workers = $300 \div 12 = 25$ workers - **Answer:** 25 workers
- Direct proportion: x/y is constant.
- Both quantities grow or shrink together.
- Direct proportion: \(y = kx\) (graph is a straight line through origin)
- Unitary method: find value of 1 unit first
- In proportion \(a : b = c : d\), \(a \times d = b \times c\)
- Extremes: first and last terms; Means: middle terms
- Direct proportion: increase together; Inverse proportion: one increases, other decreases
Inverse Proportion
In inverse proportion, one quantity increases as the other decreases, so their product x × y stays constant. With more workers a job takes fewer days: 4 workers take 6 days, so 8 workers take 3 days (4 × 6 = 8 × 3 = 24).
- Inverse proportion: x × y is constant.
- One quantity rises as the other falls.
Word Problems
What is a Scale Drawing?
A scale drawing is a drawing that represents a real object with all lengths reduced or enlarged proportionally. The scale is the ratio of the length in the drawing to the actual length.
Scale Notation:
- \(1 : 50\) means 1 unit in the drawing represents 50 units in real life
- \(1 \text{ cm} : 5 \text{ m}\) means 1 cm in drawing = 5 m in reality (units differ)
Types of Scales:
| Type | Example | Meaning |
|---|---|---|
| Reduction scale | \(1 : 100\) | Drawing is smaller than real object (maps, floor plans) |
| Enlargement scale | \(5 : 1\) | Drawing is larger than real object (microscope images) |
Using Scale Drawings:
- Actual length = Drawing length × Scale factor
- Drawing length = Actual length ÷ Scale factor
Real-Life Applications:
- Maps: 1 cm = 1 km, etc.
- Architectural blueprints: 1 cm = 1 m
- Model making: Model cars, airplanes
- Engineering drawings: Machine parts
Scale Factor:
The number used to multiply the drawing dimensions to get actual dimensions.
A map has a scale of \(1 : 50,000\). What actual distance does 8 cm on the map represent?
- Scale means 1 cm on map = 50,000 cm in reality
- 8 cm on map = \(8 \times 50,000 = 400,000\) cm
- Convert to km: \(400,000 \div 100,000 = 4\) km
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Answer: 4 km **Example 2:** In a scale drawing, a wall is 12 cm long. The scale is $1 : 50$. What is the actual length of the wall in meters? *Solution:* - Actual length = Drawing length × Scale factor - $12 \times 50 = 600$ cm - Convert to meters: $600 \div 100 = 6$ m - **Answer:** 6 m **Example 3:** The actual length of a car is 4.8 m. In a scale drawing, it is 8 cm long. Find the scale ratio. *Solution:* - Convert actual length to same unit as drawing: $4.8 \text{ m} = 480 \text{ cm}$ - Scale ratio = Drawing length : Actual length = $8 : 480$ - Simplify: Divide both by 8 → $1 : 60$ - **Answer:** $1 : 60$
- Direct: x/y constant; inverse: x × y constant.
- More people/speed → less time (inverse).
- A scale drawing represents a real object with proportional lengths
- Scale = drawing length : actual length
- For reduction: scale factor < 1 (e.g., 1:100)
- For enlargement: scale factor > 1 (e.g., 50:1)
- Always convert to same units before working with scales
- Use proportion to find actual or drawing lengths: \(\frac{\text{drawing}}{\text{actual}} = \text{scale}\)