Solved Exercises on Proportional Equations in Grade 7

Master proportional equations: ratios, cross multiplication, direct proportion, inverse proportion through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Proportion
Exercise 1
Solve the proportion:
\(\frac{x}{5} = \frac{12}{15}\)
Definition:

Proportional equation: Two ratios are equal: \(\frac{a}{b} = \frac{c}{d}\)

Cross multiplication method:
  1. Multiply diagonally: \(a \times d = b \times c\)
  2. Solve for the unknown variable
  3. Check the solution
Original Equation
\(\frac{x}{5} = \frac{12}{15}\)
Cross Multiply
\(x \times 15 = 5 \times 12\)
Solve
\(x = 4\)
Step 1: Apply cross multiplication

\(\frac{x}{5} = \frac{12}{15}\) becomes \(x \times 15 = 5 \times 12\)

Step 2: Calculate the products

\(15x = 60\)

Step 3: Solve for x

\(x = \frac{60}{15} = 4\)

Step 4: Check the solution

\(\frac{4}{5} = \frac{12}{15}\) → \(\frac{4}{5} = \frac{4}{5}\) ✓

\(x = 4\)
Final answer:

\(x = 4\)

Applied rules:

• Cross multiplication: In \(\frac{a}{b} = \frac{c}{d}\), \(ad = bc\)

• Equality: Both sides must be equal after solving

• Verification: Substitute back to confirm correctness

2 Word Problem
Exercise 2
If 3 apples cost $1.50, how much would 8 apples cost?
Definition:

Direct proportion: As one quantity increases, the other increases at the same rate

Set up proportion
\(\frac{3}{1.50} = \frac{8}{x}\)
Cross multiply
\(3x = 8 \times 1.50\)
Solve
\(x = 4.00\)
Step 1: Identify the relationship

More apples → More cost (direct proportion)

Step 2: Set up the proportion

\(\frac{\text{apples}_1}{\text{cost}_1} = \frac{\text{apples}_2}{\text{cost}_2}\)

\(\frac{3}{1.50} = \frac{8}{x}\)

Step 3: Cross multiply

\(3x = 8 \times 1.50 = 12\)

Step 4: Solve for x

\(x = \frac{12}{3} = 4.00\)

Step 5: State the answer

8 apples cost $4.00

8 apples cost $4.00
Final answer:

8 apples cost $4.00

Applied rules:

• Direct proportion: Ratios remain constant

• Unit conversion: Same units on both sides of proportion

• Problem solving: Identify the relationship first

3 Complex Proportion
Exercise 3
Solve: \(\frac{2x + 1}{3} = \frac{x - 2}{4}\)
Definition:

Complex proportion: Variables appear in both numerator and denominator

Original equation
\(\frac{2x + 1}{3} = \frac{x - 2}{4}\)
Cross multiply
\(4(2x + 1) = 3(x - 2)\)
Expand
\(8x + 4 = 3x - 6\)
Solve
\(x = -2\)
Step 1: Cross multiply

\(\frac{2x + 1}{3} = \frac{x - 2}{4}\) becomes \(4(2x + 1) = 3(x - 2)\)

Step 2: Distribute on both sides

Left side: \(4(2x + 1) = 8x + 4\)

Right side: \(3(x - 2) = 3x - 6\)

Step 3: Collect like terms

\(8x + 4 = 3x - 6\)

\(8x - 3x = -6 - 4\)

\(5x = -10\)

Step 4: Solve for x

\(x = \frac{-10}{5} = -2\)

Step 5: Verify the solution

Substitute \(x = -2\) into original equation

Left: \(\frac{2(-2) + 1}{3} = \frac{-3}{3} = -1\)

Right: \(\frac{-2 - 2}{4} = \frac{-4}{4} = -1\)

Both sides equal -1 ✓

\(x = -2\)
Final answer:

\(x = -2\)

Applied rules:

• Cross multiplication: Still applies to complex proportions

• Distribution: Distribute coefficients to all terms

• Verification: Always substitute back to check

Rules and methods, laws,...
\(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)
Cross Multiplication
Direct Proportion
\(\frac{y_1}{x_1} = \frac{y_2}{x_2}\)
As x increases, y increases
Inverse Proportion
\(x_1 \cdot y_1 = x_2 \cdot y_2\)
As x increases, y decreases
Ratio Definition
\(a:b = \frac{a}{b}\)
Comparison of quantities
Key definitions:

Ratio: Comparison of two quantities using division

Proportion: Statement that two ratios are equal

Direct Proportion: When one quantity increases, the other increases proportionally

Inverse Proportion: When one quantity increases, the other decreases proportionally

Proportion solving methods:
  1. Cross multiplication: For equations like \(\frac{a}{b} = \frac{c}{d}\), multiply \(ad = bc\)
  2. Equivalent fractions: Find common denominators
  3. Unit rate method: Find the rate per unit first
  4. Scale factor: Determine the multiplier between ratios
Tip 1: Always identify whether the relationship is direct or inverse proportion.
Tip 2: Cross multiplication is the most efficient method for solving proportions.
Tip 3: Always check your answer by substituting back into the original proportion.
Tip 4: In word problems, set up ratios with like quantities on top and bottom.
Common errors: Mixing up numerators and denominators, forgetting to cross multiply correctly, not checking units.
Exam preparation: Practice cross multiplication, word problems, and identifying direct vs inverse proportion.
Formulas to know by heart:

• Cross multiplication: \(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)

• Direct proportion: \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\)

• Inverse proportion: \(x_1 \cdot y_1 = x_2 \cdot y_2\)

• Ratio comparison: \(a:b = c:d\) means \(\frac{a}{b} = \frac{c}{d}\)

Solution: Exercises 4 to 5
4 Work Rate Problem
Exercise 4
If 4 workers can complete a job in 6 days, how many days would it take 3 workers to complete the same job?
Definition:

Inverse proportion: More workers → Fewer days (same amount of work)

Identify relationship
Workers × Days = Constant
Set up equation
\(4 \times 6 = 3 \times x\)
Solve
\(x = 8\)
Step 1: Identify the relationship

This is an inverse proportion because more workers mean fewer days to complete the same job

Step 2: Set up the inverse proportion

For inverse proportion: \(w_1 \cdot d_1 = w_2 \cdot d_2\)

Where \(w\) = workers and \(d\) = days

Step 3: Substitute known values

\(4 \times 6 = 3 \times x\)

\(24 = 3x\)

Step 4: Solve for x

\(x = \frac{24}{3} = 8\)

Step 5: Interpret the result

It would take 3 workers 8 days to complete the job

3 workers take 8 days
Final answer:

3 workers would take 8 days to complete the job

Applied rules:

• Inverse proportion: \(x_1 \cdot y_1 = x_2 \cdot y_2\)

• Work rate: Total work = Workers × Time

• Relationship identification: More workers → Less time

5 Scale Drawing Problem
Exercise 5
On a map with scale 1:50,000, if two cities are 8 cm apart, what is the actual distance between them?
Definition:

Scale ratio: Comparison between drawing size and actual size

Scale ratio
1:50,000
Set up proportion
\(\frac{1}{50,000} = \frac{8}{x}\)
Solve
\(x = 400,000\) cm
Step 1: Understand the scale

Scale 1:50,000 means 1 cm on the map represents 50,000 cm in reality

Step 2: Set up the proportion

\(\frac{\text{map distance}}{\text{actual distance}} = \frac{1}{50,000}\)

\(\frac{8}{x} = \frac{1}{50,000}\)

Step 3: Cross multiply

\(8 \times 50,000 = 1 \times x\)

\(400,000 = x\)

Step 4: Convert units

400,000 cm = 4,000 m = 4 km

Step 5: State the answer

The actual distance is 4 km

Actual distance = 4 km
Final answer:

The actual distance between the cities is 4 km

Applied rules:

• Scale interpretation: Understand what the ratio means

• Unit conversion: Convert to appropriate measurement units

• Proportion setup: Map distance over actual distance equals scale ratio

Key Concepts: Laws, Methods, Rules, Definitions
\(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)
Cross Multiplication Rule
Key definitions:

Ratio: Comparison of two quantities using division (\(a:b\) or \(\frac{a}{b}\))

Proportion: Statement that two ratios are equal (\(\frac{a}{b} = \frac{c}{d}\))

Direct Proportion: When one quantity increases, the other increases at the same rate

Inverse Proportion: When one quantity increases, the other decreases at the same rate

Scale: Ratio comparing model size to actual size

Complete methodology:
  1. Identify the type: Determine if it's direct or inverse proportion
  2. Set up the proportion: Write the ratio equation
  3. Apply the method: Cross multiplication for direct, product equality for inverse
  4. Solve the equation: Find the unknown variable
  5. Check and interpret: Verify the answer makes sense in context
Tip 1: In direct proportion, ratios are equal: \(\frac{a_1}{b_1} = \frac{a_2}{b_2}\)
Tip 2: In inverse proportion, products are equal: \(a_1 \cdot b_1 = a_2 \cdot b_2\)
Tip 3: Always keep the same units on top and bottom of each fraction
Tip 4: Cross multiplication works for any proportion: \(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)
Common errors: Mixing up direct and inverse proportion, incorrect cross multiplication, forgetting to convert units.
Exam preparation: Practice word problems, learn to identify relationship types, master cross multiplication.
Formulas to know by heart:

• Cross multiplication: \(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)

• Direct proportion: \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\)

• Inverse proportion: \(x_1 \cdot y_1 = x_2 \cdot y_2\)

• Unit rate: \(\frac{\text{quantity}}{\text{unit}}\)

• Scale: \(\frac{\text{model}}{\text{actual}} = \text{scale ratio}\)

Exercise with Visualization: Proportional Relationships
Exercise 6: Graphing Proportional Relationships
Consider the following proportional relationships:
\(y = 2x\) (Direct proportion)
\(xy = 12\) (Inverse proportion)
\(y = \frac{3}{4}x\) (Direct proportion)

Analysis: The chart shows different types of proportional relationships.

  • \(y = 2x\): Linear direct proportion passing through origin
  • \(xy = 12\): Hyperbolic inverse proportion
  • \(y = \frac{3}{4}x\): Another direct proportion with different constant

Questions & Answers

Question: I don't understand when to use direct proportion versus inverse proportion. How do I know which one to use in word problems?

Answer: Great question! The key is to think about how the quantities change relative to each other:

  • Direct proportion: "More A means more B" or "Less A means less B"
  • Inverse proportion: "More A means less B" or "Less A means more B"

Examples:

  • Direct: More hours worked → More money earned
  • Direct: More speed → More distance (in same time)
  • Inverse: More workers → Less time to finish job
  • Inverse: More speed → Less time to travel same distance

In your problems, ask yourself: "If one quantity increases, what happens to the other?" If they increase together, it's direct. If one goes up while the other goes down, it's inverse.

Question: When solving \(\frac{2x + 1}{3} = \frac{x - 2}{4}\), I sometimes forget to distribute properly. Any tips to avoid this mistake?

Answer: Distribution errors are very common! Here are strategies to avoid them:

  • Write out every step: Don't try to do distribution mentally
  • Box method: Draw boxes around each term when distributing
  • Check signs: Pay special attention to minus signs in parentheses
  • Verify with substitution: Always check your answer by plugging it back

For your example:

Left side: \(4(2x + 1) = 4 \cdot 2x + 4 \cdot 1 = 8x + 4\)

Right side: \(3(x - 2) = 3 \cdot x - 3 \cdot 2 = 3x - 6\)

Notice how the minus sign affects the 2 when distributing the 3!

Question: I have trouble with scale problems. How do I make sure I set up the proportion correctly?

Answer: Scale problems require careful attention to the order of ratios. Here's a foolproof method:

  1. Identify what the scale means: 1:50,000 means 1 unit on map = 50,000 units in reality
  2. Set up consistent ratios: Always match like quantities
  3. Write the template: \(\frac{\text{map measurement}}{\text{actual measurement}} = \frac{\text{scale map}}{\text{scale actual}}\)

For a scale of 1:50,000:

\(\frac{\text{distance on map}}{\text{actual distance}} = \frac{1}{50,000}\)

If the map distance is 8 cm:

\(\frac{8}{x} = \frac{1}{50,000}\)

This ensures the ratios are properly aligned!