Proportion, Ratio and Drawing Scale

Definition and rules

• The ratio of two quantities of the same kind and same units is a : b = a/b; it is a pure number, it has no unit.
• Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio, so 12 : 18 = 2 : 3.
• The ratio is always written in its lowest terms, and the order of the terms matters: the ratio of a to b is not the ratio of b to a.

Dividing a quantity in a given ratio

To divide Q in the ratio a : b : c, find the total number of parts a + b + c, then the share of each part is Q ÷ (a + b + c); multiply by a, b, c.

• Divide 240 feddans in the ratio 3 : 5. Parts = 8, one part = 30, so the shares are 90 and 150 feddans.
• Divide EGP 1,200 in the ratio 2 : 3 : 7. Parts = 12, one part = 100, so 200, 300, 700.

Ratio problems with two changing quantities

• The ratio of the ages of two friends is 4 : 5 and the difference of their ages is 6 years. One part = 6, so their ages are 24 and 30 years.
• If a : b = 2 : 3 and b : c = 4 : 5, make b common: a : b : c = 8 : 12 : 15.

Definition

Four quantities a, b, c, d are in proportion when a : b = c : d, written a : b :: c : d. Then a and d are the extremes, b and c are the means, and the fundamental property is:

• product of extremes = product of means ⟹ a × d = b × c.
• From a : b = c : d we may also write b : a = d : c, a : c = b : d and (a + b) : b = (c + d) : d.

Standard terms

• Fourth proportional to a, b, c: the number x with a : b = c : x, so x = b c / a. For 2, 5, 8 → x = 5 × 8 / 2 = 20.
• Third proportional to a, b: the number x with a : b = b : x, so x = b²/a. For 4, 6 → x = 36/4 = 9.
• Mean proportional between a and b: the number x with a : x = x : b, so x = √(a b). Between 4 and 9 → x = 6.

Continued proportion

Three quantities are in continued proportion when a : b = b : c, i.e. b² = a c; then b is the mean proportional and a : c = a² : b².

• If a : b = 2 : 3 and b : c = 3 : 7, then a : b : c = 2 : 3 : 7 and a : c = 2 : 7.

Direct proportion

Two quantities are in direct proportion when they increase (or decrease) together at a constant ratio: x ∝ y ⟹ x = k y, k constant. The graph is a straight line through the origin and its slope is k.

• Cost ∝ number of metres of cloth. If 15 m costs EGP 450, the constant is 30 EGP/m, so 24 m cost 24 × 30 = EGP 720.
• At constant speed, distance ∝ time; at constant width, the area of a rectangle ∝ its length.

Inverse proportion

Two quantities are in inverse proportion when one increases as the other decreases so that their product stays constant: x ∝ 1/y ⟹ x y = k. The graph is a hyperbola.

• For a fixed distance, speed × time = constant. A car covers a distance in 3 h at 80 km/h; at 60 km/h the time = 240/60 = 4 h.
• Work problem: number of workers × number of days = constant for the same job. 12 workers finish in 20 days, so 15 workers finish in 12 × 20 / 15 = 16 days.

Choosing between them

Ask: when one quantity doubles, does the other double (direct) or halve (inverse)? Mixing the two is the most common error in this topic.

Definition

The scale of a drawing, map or plan is the ratio:

• scale = length on the drawing : the corresponding real length, both in the same units.
• A scale of 1 : 50,000 means 1 cm on the map represents 50,000 cm = 500 m = 0.5 km on the ground.
• A scale of 1 : 100 means 1 cm on the plan = 100 cm = 1 m in reality.
• An enlarging scale (a machine part, a bacterium) is written the other way, e.g. 20 : 1 — drawing : real.

The three conversions

1. Real length = drawing length × the second term (for a 1 : n scale). A field measures 4 cm × 6 cm on a plan of scale 1 : 2,000 ⟹ real sides 80 m and 120 m.
2. Drawing length = real length ÷ n. A road 3.6 km long = 360,000 cm appears on a 1 : 20,000 map as 360,000 ÷ 20,000 = 18 cm.
3. Scale = drawing ÷ real after converting the real length to the same unit.

Area and volume on a scaled drawing

• Lengths are multiplied by n, so areas by n² and volumes by n³.
• A plot drawn as 4 cm × 6 cm at scale 1 : 2,000 has real area 80 × 120 = 9,600 m², and the drawing area is 24 cm² = 9,600 m² ÷ (2,000)² × 10,000 — check: 24 cm² × 2000² = 96,000,000 cm² = 9,600 m².
• So: real area = area on plan × n², and 1 feddan = 4,200 m², 1 km² = 100 hectares = 238 feddans approximately.

Worked map example

Two cities are 7 cm apart on a map of scale 1 : 250,000. The real distance = 7 × 250,000 = 1,750,000 cm = 17.5 km. A train covering it in 15 minutes travels 17.5 ÷ 0.25 = 70 km/h.

1. Express in lowest terms: 36 : 48, 2.5 : 7.5, 150 p : 1 EGP (1 EGP = 100 p). — 3 : 4, 1 : 3, 3 : 2 after converting to the same unit.
2. Divide EGP 2,700 between two people in the ratio 4 : 5. — 1,200 and 1,500 (parts 9, one part 300).
3. The angles of a triangle are in the ratio 2 : 3 : 4. Find them. — Parts 9, one part = 180/9 = 20° ⟹ 40°, 60°, 80°.
4. Find the fourth proportional to 3, 7, 12. — x = 7 × 12 / 3 = 28.
5. Find the mean proportional between 8 and 18. — √144 = 12.
6. If x ∝ y and x = 10 when y = 4, find x when y = 7. — k = 2.5 ⟹ x = 17.5.
7. A job takes 9 days at 8 hours a day. How many days at 6 hours a day? — 72 hours of work ⟹ 12 days.
8. A city is 5 cm from another on a map of scale 1 : 400,000. Find the real distance in km. — 2,000,000 cm = 20 km.
9. A rectangular field 90 m × 60 m is drawn at scale 1 : 1,500. Find its dimensions on the plan. — 9,000 ÷ 1,500 = 6 cm and 6,000 ÷ 1,500 = 4 cm.
10. A plan of scale 1 : 1,000 shows a garden of area 12 cm². Find the real area in m². — 12 × 1,000² = 12,000,000 cm² = 1,200 m².
11. A machine part 4 mm long is drawn 8 cm. Find the scale. — 80 : 4 = 20 : 1 (an enlarging scale).