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Text lesson

Proportion, Ratio and Drawing Scale

Mathematics

Proportion, Ratio and Drawing Scale

Overview

Overview

  • Compare quantities by ratio and divide an amount in a given ratio.
  • Form and solve a proportion, and tell direct from inverse proportion.
  • Use a drawing scale to convert between a plan (or map) length and the real length.
  • Relate lengths, areas and volumes on a scaled drawing to the real ones.

🌳 Concept tree

Ratio & Proportion

├── Ratio

│ ├── a→b, same units only

│ ├── Equivalent ratios (× or ÷ both terms)

│ └── Dividing a quantity in the ratio a : b

├── Proportion

Comparison aspectMain ideaKeywords
│ ├── ab = c : d ⟹ a d = b c
•│ ├── Fourth proportional / third proportional / mean proportionalFourthproportionalthirdmean
│ └── Continued proportion ab = b : c
├── Variation
│ ├── Directx ∝ y ⟹ x = k y
│ └── Inversex ∝ 1/y ⟹ x y = k

└── Drawing scale

Comparison aspectMain ideaKeywords
├── scale = drawing lengthreal lengthreal
├── 1n ⟹ real = drawing × nrealdrawing
•
  • └── area on plan = real area ÷ n2
  • volume = real ÷ n3
areaplanrealvolume
Proportion, Ratio and Drawing Scale · 1 / 7
Ratio

Ratio

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 35. Parts = 8, one part = 30, so the shares are 90 and 150 feddans.
Divide EGP 1,200 in the ratio 23 : 7. Parts = 12, one part = 100, so 200, 300, 700.
The ratio of the ages of two friends is 45 and the difference of their ages is 6 years. One part = 6, so their ages are 24 and 30 years.
If ab = 2 : 3 and b : c = 4 : 5, make b common: a : b : c = 8 : 12 : 15.
Proportion, Ratio and Drawing Scale · 2 / 7
Proportion

Proportion

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 ab = c : d we may also write b : a = d : c, a : c = b : d and (a + b) : b = (c + d) : d.
Fourth proportionalto 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 proportionalto a, b: the number x with a : b = b : x, so x = b2/a. For 4, 6 → x = 36/4 = 9.
Mean proportionalbetween 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.
Proportion, Ratio and Drawing Scale · 3 / 7
Direct and inverse proportion

Direct and inverse proportion

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.
Proportion, Ratio and Drawing Scale · 4 / 7
Drawing scale (مقياس الرسم)

Drawing scale (مقياس الرسم)

Definition

  1. The scale of a drawing
  2. map or plan is the ratio:
Proportion, Ratio and Drawing Scale: a wooden scale model of a house at drawing scale 1 : 50: 20 cm on the model is 10 m in reality.
Proportion, Ratio and Drawing Scale: a wooden scale model of a house at drawing scale 1 : 50: 20 cm on the model is 10 m in reality.
Comparison aspectMain ideaExampleKeywords
scale = length on the drawing : the corresponding real lengthboth in the same units—bothsameunits
A scale of 1
  • 000 means 1 cm on the map represents 50
  • 000 cm = 500 m = 0.5 km on the ground
—meansrepresentsground
A scale of 1100 means 1 cm on the plan = 100 cm = 1 m in reality—meansplanreality
•
  • An enlarging scale (a machine part
  • a bacterium) is written the other way
20 : 1 — drawing : realenlargingscalemachinepart
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.
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.
Scale = drawing ÷ realafter converting the real length to the same unit.

Area and volume on a scaled drawing

Comparison aspectMain ideaKeywords
•
  • Lengths are multiplied by n
  • so areas by n2 and volumes by n3
Lengthsmultipliedareasvolumes
A plot drawn as 4 cm × 6 cm at scale 1
  • 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
realareadrawingcheck
So
  • real area = area on plan × n2
  • and 1 feddan = 4
  • 200 m2
  • 1 km2 = 100 hectares = 238 feddans approximately
realareaplanfeddan

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.
Proportion, Ratio and Drawing Scale · 5 / 7
Practice with answers

Practice with answers

  1. 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. 5. — 1,200 and 1,500 (parts 9, one part 300).
  3. 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. 400,000. Find the real distance in km. — 2,000,000 cm = 20 km.
  9. 1,500. Find its dimensions on the plan. — 9,000 ÷ 1,500 = 6 cm and 6,000 ÷ 1,500 = 4 cm.
  10. 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).
Proportion, Ratio and Drawing Scale · 6 / 7
Exam traps

Exam traps

Comparison aspectMain ideaKeywords
Different units.
  • Never write a ratio of cm to m
  • convert first. 150 piastres to 1 EGP is 3 : 2
  • not 150 : 1
Neverwriteratioconvert
Order of the terms.
  • “The ratio of boys to girls is 3 : 5” means 3 shares boys
  • 5 shares girls
  • reversing them loses the marks in the same question
ratioboysgirlsmeans
Area is n2, not n.
  • The classic multiple-choice trap: a plan at 1 : 1
  • 000 with 12 cm2 gives 1
  • 200 m2 (n2)
  • not 12
  • 000 m2 (n)
classicmultiplechoicetrap
Enlarging versus reducing scales.
  • Maps use 1 : n
  • a drawing of a small part uses n : 1. Check
  • term is 1
Mapsdrawingsmallpart
Direct versus inverse.
  • In work problems the number of workers is inversely proportional to the number of days
  • in cost problems it is directly proportional to the quantity
workproblemsnumberworkers
Continued proportion.
  • a : b = b : c gives b2 = a c
  • students often write b2 = a + c
givesstudentsoftenwrite
Proportion, Ratio and Drawing Scale · 7 / 7