Ch 3: Proportional Reasoning-2 Quick revision notes Class 8th Mathematics (Ganita Manjari-II)

Quick revision notes

Ch 3: Proportional Reasoning-2 Quick revision notes Class 8th Mathematics (Ganita Manjari-II)

Class 8 · Mathematics (Ganita Prakash) · Chapter 3 : Proportional Reasoning-2 · All Board · ENGLISH · 13 views

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⚖️

Section 3.1

Proportionality — A Quick Recap

📌 What is a Proportional Relationship?

When two or more quantities change by the same factor, we call that a proportional relationship.

Example: Idli batter uses rice and urad dal in ratio $2 : 1$ — for every 2 cups of rice, 1 cup of urad dal.

✍️ Ratio Notation

Two quantities in proportion written as $a : b$

Rice : Urad Dal = $2 : 1$

✅ Cross Multiplication Test

Two ratios $a:b$ and $c:d$ are proportional if:

\[ a \times d = b \times c \]

Cross multiplication diagram 6:3 and 4:2

Cross-multiplication check: 6×2 = 12 = 3×4 ✔ — ratios 6:3 and 4:2 are proportional

General Rule Ratios $a : b$ and $c : d$ are proportional when:

\[ a \times d = b \times c \quad \Leftrightarrow \quad \frac{a}{c} = \frac{b}{d} \]

💡

Key Point: Proportional does NOT mean equal quantities — it means equal ratios.
6:3 and 4:2 both simplify to 2:1.

🗺️

Section 3.2

Ratios in Maps

Map of South India — RF = 1 : 60,00,000 shown in the lower right corner

📐 Representative Fraction (RF)

The ratio shown on a map that tells us:

\[ \text{RF} = \frac{\text{Distance on map}}{\text{Actual distance on ground}} \]

🔢 Reading RF = 1 : 60,00,000

1 cm on map = 60,00,000 cm on ground

\[ 60{,}00{,}000 \text{ cm} = 60 \text{ km} \]

📏 Formula

\[ \text{Actual Dist.} = \text{Map Dist.} \times \text{RF denominator} \]

🔍 Steps to Find Geographical Distance

1
Measure distance between two cities on the map using a ruler (get value in cm)
2
Multiply map distance (cm) by the RF denominator → get actual distance in cm
3
Divide by 1,00,000 to convert cm → km

⚠️

Important: RF gives geographical (straight-line) distance, NOT road distance!

🌶️

Section 3.3

Ratios with More than 2 Terms

Many Ingredients, One Ratio

Ratios can have as many terms as needed — one for each quantity — provided all change by the same factor.

Ingredient	Amount
Coriander seeds	8 spoons
Red chillies	4
Toor dal	2 spoons
Fenugreek (methi)	1 spoon

Viswanath's Spice Ratio \[ 8 : 4 : 2 : 1 \]

Mixing spices — proportional ratio keeps the flavour same

📐 General Rule — Two Multi-term Ratios are Proportional When:

\[ a:b:c:d \;\;::\;\; p:q:r:s \quad \Rightarrow \quad \frac{a}{p} = \frac{b}{q} = \frac{c}{r} = \frac{d}{s} \]

🔄

Scaling a multi-term ratio: Multiply or divide all terms by the same number.

\[ 8:4:2:1 \xrightarrow{\div 2} 4:2:1:0.5 \]

So $8:4:2:1 \;\;::\;\; 4:2:1:0.5$

🎨 Purple Paint — Example 1

Red : Blue : White $= 2:3:5$

White = 10 L ↔ 5 parts

\[ 1 \text{ part} = \frac{10}{5} = 2 \text{ L} \]

Red $= 2 \times 2 =$ 4 L

Blue $= 3 \times 2 =$ 6 L

🏗️ Concrete Mix — Example 2

Cement : Sand : Gravel $= 1:1.5:3$

With 3 bags cement → multiply all by 3

\[ 3:4.5:9 \]

Total = $3+4.5+9=$ 16.5 bags

➗

Section 3.4

Dividing a Whole in a Given Ratio

📌 The Master Formula

To divide quantity $x$ in ratio $a : b : c : \ldots$

\[ \text{1st part} = x \times \frac{a}{a+b+c+\cdots} \quad \text{2nd part} = x \times \frac{b}{a+b+c+\cdots} \quad \ldots \]

Concrete mixture diagram — 110 units split into cement sand gravel

110 units of concrete split in ratio 1 : 1.5 : 3 → 20 cement + 30 sand + 60 gravel (Example 3)

🎨 Example 4 — 50 ml Purple Paint

Ratio $2:3:5$, Total parts $= 10$

\[ \text{Red} = 50 \times \tfrac{2}{10} = 10 \text{ ml} \] \[ \text{Blue} = 50 \times \tfrac{3}{10} = 15 \text{ ml} \] \[ \text{White} = 50 \times \tfrac{5}{10} = 25 \text{ ml} \]

📐 Example 5 — Triangle Angles in 1:3:5

Sum of angles $= 180°$, Total parts $= 9$

\[ \angle A = 180° \times \tfrac{1}{9} = 20° \] \[ \angle B = 180° \times \tfrac{3}{9} = 60° \] \[ \angle C = 180° \times \tfrac{5}{9} = 100° \]

Triangle with angles in ratio 1:3:5 → 20°, 60°, 100°

🥧

Section 3.5

A Slice of the Pie — Pie Charts

What is a Pie Chart?

A circular chart where each slice angle is proportional to the quantity it represents. Total = 360°.

Slice Angle Formula

\[ \text{Angle} = \frac{\text{Category Value}}{\text{Total}} \times 360° \]

💡

Tip: Simplify the ratio first by dividing all values by their HCF.

$Coloured pie chart — Grade A B C D E for 40 students$

Grade distribution pie chart — 40 students

📊 Grade Distribution Worked Example

Students: A=12, B=10, C=8, D=6, E=4. Total = 40. HCF = 2 → Ratio = $6:5:4:3:2$ , Sum = 20.

\[ \text{Grade A} = \frac{6}{20} \times 360° = 6 \times 18 = 108° \quad \text{Grade B} = \frac{5}{20} \times 360° = 90° \] \[ \text{Grade C} = 4 \times 18 = 72° \quad \text{Grade D} = 3 \times 18 = 54° \quad \text{Grade E} = 2 \times 18 = 36° \]

Step 1: Draw circle with radius AB | Step 2: Measure 108° anti-clockwise for Grade A

Pie chart construction steps 3 to 7 — complete coloured chart

Steps 3–7: Add all slices (90°, 72°, 54°, 36°) → Colour and label the completed pie chart

🖊️ How to Draw a Pie Chart

1
Calculate angle for each category: $\text{angle} = \frac{\text{value}}{\text{total}} \times 360°$
2
Draw a circle and mark starting radius AB
3
Use protractor — measure each angle from the last radius, draw new radius
4
Continue for all slices (the last slice fills the remaining space automatically)
5
Colour and label each slice with its category name

🔄

Section 3.6

Inverse Proportions

Direct Proportion

Both quantities increase or decrease together

\[ \frac{x_1}{y_1} = \frac{x_2}{y_2} = k \]

Example: more workers → more bricks moved

Inverse Proportion

One increases → other decreases by same factor

\[ x_1 \cdot y_1 = x_2 \cdot y_2 = k \]

Example: more speed → less time taken

🔑 Definition

Two quantities $x$ and $y$ are in inverse proportion if their product is always constant:

\[ x \times y = k \quad (\text{constant}) \]

Also: if $x$ changes by factor $n$, then $y$ changes by factor $\dfrac{1}{n}$.

🚗 Speed–Time Example

Mode	Speed (km/h)	Time (hr)
Walk	5	18
Bicycle	15	6
Motorcycle	30	3
Car	60	1.5

\[ 5 \times 18 = 15 \times 6 = 30 \times 3 = 60 \times 1.5 = 90 \]

Product (= distance) is always 90 km ✔

Motorcycle and car — speed vs time illustration

More speed → less time: inverse proportion

Inverse proportion table with multiplication factor arrows

Speed multiplies by ×3 → Time divides by ×⅓ (same factor, opposite direction)

Key Formula

\[ x_1 y_1 = x_2 y_2 = k \quad \Leftrightarrow \quad \frac{x_1}{x_2} = \frac{y_2}{y_1} \]

🧱 Example 3 — Workers & Road

20 workers → 4 days (fewer workers = more days)

\[ 20 \times 4 = 10 \times y_2 \Rightarrow y_2 = \frac{80}{10} = \mathbf{8 \text{ days}} \]

💧 Example 4 — Pumps & Tank

2 pumps → 18 hr (more pumps = less time)

\[ 2 \times 18 = 4 \times x \Rightarrow x = \frac{36}{4} = \mathbf{9 \text{ hr}} \]

🍱 Example 5 — Food Provisions

80 students → 15 days; now 100 students

\[ 80 \times 15 = 100 \times x \Rightarrow x = \frac{1200}{100} = \mathbf{12 \text{ days}} \]

🥕 Example 6 — Working Together

Ram: 1 hr alone; Shyam: 1.5 hr alone

\[ \text{Combined rate} = 1 + \frac{1}{1.5} = 1 + \frac{2}{3} = \frac{5}{3} \text{ units/hr} \] \[ \text{Time together} = \frac{1}{\frac{5}{3}} = \frac{3}{5} \text{ hr} \]

🔍 Quick Test: Is it Inverse Proportion?

Check if product $x \times y$ is the same for all pairs of values. If yes → Inverse Proportion ✓

Check if ratio $x/y$ is constant → Direct Proportion ✓

⭐ Quick Summary

Ratios $a:b$ and $c:d$ are proportional when $a \times d = b \times c$

RF on a map = map distance : actual ground distance. RF 1:60,00,000 means 1 cm = 60 km (geographical, not road).

Multi-term ratios $a:b:c:d \;\;::\;\; p:q:r:s$ are proportional when $\dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r} = \dfrac{d}{s}$

To divide $x$ in ratio $a:b:c$: each share $= x \times \dfrac{\text{term}}{a+b+c}$

Pie chart: angle of each slice $= \dfrac{\text{value}}{\text{total}} \times 360°$

Direct proportion: $\dfrac{x_1}{y_1} = \dfrac{x_2}{y_2} = k$ (ratio constant, same direction)

Inverse proportion: $x_1 y_1 = x_2 y_2 = k$ (product constant, opposite direction)

Notes by @edugrown · Ganita Prakash Grade 8 · Chapter 3 · Proportional Reasoning 2