Ratio, Proportion and Rate

Key Concepts

• Ratio

  • A ratio compares two quantities of the same kind.
  • It shows how much of one quantity there is compared to another.
  • Ratios can be written in 3 ways:
    • a : b
    • a/b
    • “a to b”
  • Example: If there are 2 boys and 3 girls, the ratio of boys to girls is 2 : 3.

• Equivalent ratios

  • Two ratios are equivalent if they represent the same comparison.
  • Example: 2 : 3 = 4 : 6 = 6 : 9
  • You get equivalent ratios by multiplying or dividing both parts by the same number.

• Proportion

  • Proportion means two ratios are equal.
  • Example: 2/3 = 4/6
  • In proportion questions, you often find a missing value by using the fact that the ratios are equal.

• Direct proportion

  • Two quantities are in direct proportion if:
    • when one quantity increases, the other also increases in the same ratio
    • when one quantity decreases, the other also decreases in the same ratio
  • If y is directly proportional to x, we write:
    • y ∝ x
    • y = kx
  • Here, k is a constant of proportionality.
  • Example:
    • Cost is directly proportional to number of items if each item has the same price.
    • If 1 notebook costs $2, then 3 notebooks cost $6.

• How to recognise direct proportion

  • The ratio y/x is constant.
  • This constant value is k.
  • Example:
    • If y = 3x, then y/x = 3 for all values of x.

• Graph of direct proportion

  • A graph of direct proportion is a straight line passing through the origin (0,0).
  • The equation is y = kx.
  • The steeper the line, the larger the value of k.

• Inverse proportion

  • Two quantities are in inverse proportion if:
    • when one quantity increases, the other decreases
    • their product stays constant
  • If y is inversely proportional to x, we write:
    • y ∝ 1/x
    • y = k/x
  • Here, k is also a constant of proportionality.
  • Example:
    • Time taken to finish a job is inversely proportional to the number of workers, if all workers work at the same rate.

• How to recognise inverse proportion

  • The product xy is constant.
  • This constant value is k.
  • Example:
    • If y = 12/x, then xy = 12.

• Graph of inverse proportion

  • A graph of inverse proportion is a curve, not a straight line.
  • It gets closer and closer to the axes but does not touch them.
  • If both x and y are positive, the curve lies in the first quadrant.
  • The equation is y = k/x.

• Rate

  • A rate compares two quantities measured in different units.
  • Example:
    • 60 km/h
    • 80 beats per minute
    • $5 per kg
  • The word “per” usually means division.

• Speed

  • Speed is a type of rate.
  • It tells you the distance travelled per unit time.
  • Formula:
    • Speed = Distance / Time
  • Rearranged formulas:
    • Distance = Speed × Time
    • Time = Distance / Speed

• Units of speed

  • Common units:
    • m/s
    • km/h
    • cm/s
  • You must make sure distance and time are in matching units.
  • Example:
    • km with h
    • m with s

• Map scale

  • A map scale shows the relationship between a distance on a map and the actual distance on the ground.
  • It helps us convert between map distance and real distance.

• Types of map scale

  • Statement scale: e.g. 1 cm represents 5 km
  • Ratio scale: e.g. 1 : 500 000
  • A ratio scale means:
    • 1 unit on the map represents 500 000 of the same units in real life
    • e.g. 1 cm on map = 500 000 cm in reality

• Distance scale

  • Used to compare lengths.
  • For example, if the scale is 1 : 100 000, then:
    • 1 cm on map = 100 000 cm actual
    • 100 000 cm = 1 km

• Area scale

  • Area scale is different from distance scale.
  • If the distance scale is 1 : n, then the area scale is 1 : n².
  • Example:
    • If length scale is 1 : 1000
    • Area scale is 1 : 1 000 000
  • This is because area involves two dimensions: length × width.

• Expressing relationships as equations

  • Mathematical relationships can be written using equations.
  • This helps describe how one quantity changes with another.
  • Examples:
    • Direct proportion: y = kx
    • Inverse proportion: y = k/x
    • Speed relationship: d = st
  • To form an equation:
    1. Identify the variables.
    2. Decide whether the relationship is direct, inverse, or another rule.
    3. Use the given information to find constants.
    4. Write the final equation clearly.

Important Definitions

• Ratio: a comparison of two quantities of the same kind using division.
• Equivalent ratios: ratios that have the same value when simplified.
• Proportion: a statement that two ratios are equal.
• Direct proportion: a relationship in which two quantities change in the same ratio.
• Inverse proportion: a relationship in which one quantity increases while the other decreases so that their product remains constant.
• Constant of proportionality: a fixed number, usually written as k, in proportional relationships.
• Graph: a visual representation of the relationship between two variables on axes.
• Origin: the point (0,0) on a graph where the x-axis and y-axis meet.
• Rate: a comparison of two quantities with different units.
• Speed: the distance travelled per unit time.
• Map scale: the relationship between a distance on a map and the corresponding actual distance.
• Ratio scale: a map scale written in the form 1 : n.
• Statement scale: a map scale written using words, such as 1 cm represents 2 km.
• Distance scale: a scale used to compare lengths or distances.
• Area scale: a scale used to compare areas; it is the square of the distance scale.
• Equation: a mathematical statement showing that two expressions are equal.
• Variable: a symbol, such as x or y, that represents a quantity that can change.

Worked Examples

Example 1: Direct Proportion

3 packets of stickers cost $4.50. Find the cost of 8 packets, assuming cost is directly proportional to the number of packets.

Step 1: Identify the relationship

• Cost is directly proportional to number of packets.
• Let cost be C dollars and number of packets be n.
• So, C = kn

Step 2: Use the given information to find k

• When n = 3, C = 4.50
• Substitute:
  • 4.50 = k(3)
  • k = 4.50 / 3 = 1.50

So the equation is:

• C = 1.50n

Step 3: Find the cost of 8 packets

• C = 1.50(8)
• C = 12.00

Answer:

• The cost of 8 packets is $12.00.

Example 2: Inverse Proportion

6 workers can complete a job in 15 days. How many days will 10 workers take, assuming all workers work at the same rate?

Step 1: Identify the relationship

• Number of workers and number of days are in inverse proportion.
• More workers means fewer days.
• Let number of workers be w and number of days be d.
• So, d = k/w

Step 2: Use the constant product

• For inverse proportion:
  • wd = k
• Given:
  • 6 × 15 = 90
• So k = 90

Thus:

• d = 90/w

Step 3: Find the number of days for 10 workers

• d = 90/10
• d = 9

Answer:

• 10 workers will take 9 days.

Example 3: Map Scale and Speed

A map has scale 1 : 200 000. The distance between Town A and Town B on the map is 7.5 cm. A car travels from A to B at a speed of 60 km/h. How long does the journey take?

Step 1: Convert map distance to actual distance

• Scale 1 : 200 000 means:
  • 1 cm on map = 200 000 cm in reality
• Actual distance:
  • 7.5 × 200 000 = 1 500 000 cm

Step 2: Convert to kilometres

• 100 000 cm = 1 km
• So:
  • 1 500 000 cm = 15 km

Step 3: Use the speed formula

• Time = Distance / Speed
• Time = 15 / 60
• Time = 0.25 h

Step 4: Convert hours to minutes

• 0.25 × 60 = 15

Answer:

• The journey takes 15 minutes.

Common Mistakes to Avoid

• Confusing direct proportion with inverse proportion.

  • Direct proportion: both increase or both decrease together.
  • Inverse proportion: one increases while the other decreases.

• Forgetting the correct formula:

  • Direct proportion: y = kx
  • Inverse proportion: y = k/x

• Thinking every straight-line graph shows direct proportion.

  • A direct proportion graph must be a straight line through the origin.

• Forgetting that inverse proportion graphs are curves, not straight lines.

• Using y/x to test inverse proportion.

  • For inverse proportion, test whether xy is constant.

• Not converting units before solving speed questions.

  • Example: using km with minutes without converting.

• Mixing up the formulas:

  • Speed = Distance / Time
  • not Time / Distance

• Converting map scales incorrectly.

  • In a ratio scale, both sides must be in the same unit.

• Forgetting that area scale is the square of distance scale.

  • If scale is 1 : 100, area scale is 1 : 10 000, not 1 : 100.

• Rounding too early in calculations.

  • Keep full values until the final answer unless instructed otherwise.

• Writing equations without defining variables.

  • Always say what x, y, d, t, or s represent.

Exam Tips

• Look for keywords:

  • Direct proportion: “varies directly”, “proportional to”, “same rate”
  • Inverse proportion: “varies inversely”, “more means less”, “product is constant”
  • Rate: “per”, “for each”
  • Speed: “distance”, “time”, “travel”

• When asked to form an equation:

  • Define the variables first.
  • Write the proportional relationship.
  • Find the constant k.
  • State the final equation clearly.

• For direct proportion questions, include phrases like:

  • “Since y is directly proportional to x, y = kx.”

• For inverse proportion questions, include phrases like:

  • “Since y is inversely proportional to x, y = k/x.”
  • or “xy is constant.”

• For graph questions:

  • Mention whether the graph is a straight line through the origin or a curve approaching the axes.

• For map scale questions:

  • Convert to the same units before using the scale.
  • Show the unit conversion clearly for method marks.

• For speed questions:

  • Write the formula first.
  • Substitute values with units.
  • Convert the final answer to sensible units if needed.

• If the answer is a time:

  • Check whether the question wants the answer in hours, minutes, or seconds.

• Always include units:

  • km, cm, h, min, m/s, km/h, cm², km²

• If area is involved:

  • Remember to square the scale factor.

Quick Summary

• A ratio compares two quantities of the same kind.
• Proportion means two ratios are equal.
• In direct proportion, one quantity changes in the same ratio as the other.
• Direct proportion formula: y = kx
• In direct proportion, y/x is constant.
• The graph of direct proportion is a straight line through the origin.
• In inverse proportion, one quantity increases while the other decreases.
• Inverse proportion formula: y = k/x
• In inverse proportion, xy is constant.
• The graph of inverse proportion is a curve approaching the axes.
• A rate compares two quantities with different units; speed is distance per unit time.
• Speed formulas: s = d/t, d = st, t = d/s
• A map scale links map distance to actual distance.
• For area, square the distance scale factor.
• Always convert units carefully and include them in your final answer.