Proportional Reasoning – 2

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CLASS VIII Mathematics ~4–5 marks Ch 10 of 14
Proportional Reasoning – 2

Class 8 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • Two ratios $a:b$ and $c:d$ are proportional when $a\times d=b\times c$ (cross-multiply), equivalently $\dfrac{a}{c}=\dfrac{b}{d}$.
  • A map's Representative Fraction (RF), e.g. $1:60{,}00{,}000$, is the ratio of map distance to actual ground distance — $1$ cm on the map $=60{,}00{,}000$ cm $=60$ km on the ground.
  • Ratios can have many terms, $a:b:c:d$, when all quantities scale by the same factor. To divide $x$ in ratio $p:q:r$, each part is $x\times\dfrac{\text{its term}}{\text{sum of terms}}$.
  • Pie charts split $360^\circ$ in the ratio of the data; each slice angle $=\dfrac{\text{value}}{\text{total}}\times360^\circ$.
  • Direct proportion: $\dfrac{x_1}{y_1}=\dfrac{x_2}{y_2}=k$ (quotient constant). Inverse proportion: $x_1y_1=x_2y_2=k$ (product constant).
  • Weightage: ~4–5 marks — typically one ratio-sharing / map-scale problem and one direct-or-inverse proportion word problem (including time-and-work).
Detailed notes

1. Proportionality — a quick recap

When two or more related quantities change by the same factor, we call that a proportional relationship. We write it with the ratio notation. For example, idli batter mixes rice and urad dal in a fixed proportion: for $2$ cups of rice we add $1$ cup of urad dal, written $2:1$.

Viswanath mixes $6$ cups of rice with $3$ cups of dal ($6:3$); Puneet mixes $4$ cups of rice with $2$ cups of dal ($4:2$). Would they taste the same? Check by cross-multiplication: $6\times2=12$ and $3\times4=12$. The two products are equal, so $6:3$ and $4:2$ are proportional — same taste.

$$a:b\ \text{and}\ c:d\ \text{are proportional}\iff a\times d=b\times c\iff \frac{a}{c}=\frac{b}{d}$$

This single cross-multiplication test is the backbone of the whole chapter — use it whenever you must check if two ratios match.

2. Ratios in maps — the Representative Fraction (RF)

Most maps print a ratio in a corner, like $\text{RF}=1:60{,}00{,}000$. A Representative Fraction (RF) is the ratio between a distance on the map and the actual distance on the ground.

So $1:60{,}00{,}000$ means $1$ cm on the map represents $60{,}00{,}000$ cm on the ground (the straight-line geographical distance, not the road distance).

$$60{,}00{,}000\ \text{cm}=\frac{60{,}00{,}000}{100}\ \text{m}=60{,}000\ \text{m}=60\ \text{km}$$

To find a real distance: measure the map distance with a ruler, then multiply by the RF's big number. If two cities are $4$ cm apart on a $1:60{,}00{,}000$ map, the ground distance is $4\times60=240$ km. Different maps with different scales should give roughly the same ground distance for the same two cities. A classroom sketch at scale $1:50$ means $1$ cm on paper $=50$ cm in the real room.

3. Ratios with more than two terms

A ratio can have many terms when every quantity scales by the same factor. Viswanath's spice mix grinds $8$ spoons coriander : $4$ red chillies : $2$ toor dal : $1$ fenugreek, written $8:4:2:1$ (four terms).

Puneet has only $2$ red chillies — that is half of Viswanath's $4$. To keep the same taste, halve every ingredient: $4:2:1:0.5$. So $8:4:2:1 :: 4:2:1:0.5$ (both proportional).

$$a:b:c:d :: p:q:r:s \iff \frac{a}{p}=\frac{b}{q}=\frac{c}{r}=\frac{d}{s}$$
NCERT Example 1 — purple paint Red : Blue : White $= 2:3:5$

Yasmin has $10$ L of white paint (the $5$-part term). If $5$ parts $=10$ L, then $1$ part $=10\div5=2$ L. So Red $=2\times2=4$ L, Blue $=3\times2=6$ L. Total purple paint $=4+6+10=20$ L.

NCERT Example 2 — concrete cement : sand : gravel $=1:1.5:3$

With $3$ bags of cement, multiply each term by $3$: cement : sand : gravel $=3:4.5:9$. Total concrete $=3+4.5+9=16.5$ bags.

4. Dividing a whole in a given ratio

To split a quantity in a ratio, add the terms to get the total parts, then give each part its share.

Simple two-term case: divide $12$ in $2:1$. Sum $=2+1=3$; one part $=12\div3=4$; so $2\times4=8$ and $1\times4=4$. Thus $12=8:4$.

$$\text{For ratio } a:b:c:\dots,\quad \text{first part}=x\times\frac{a}{a+b+c+\dots},\ \text{second}=x\times\frac{b}{a+b+c+\dots},\ \dots$$
NCERT Example 3 — $110$ units of concrete in $1:1.5:3$

Sum $=1+1.5+3=5.5$. We need $110\div5.5=20$ times the recipe, so multiply each term by $20$: cement $=20$, sand $=30$, gravel $=60$ units. (Check: $20+30+60=110$.)

NCERT Example 4 — $50$ ml purple paint in $2:3:5$

Sum $=2+3+5=10$. Red $=50\times\dfrac{2}{10}=10$ ml; Blue $=50\times\dfrac{3}{10}=15$ ml; White $=50\times\dfrac{5}{10}=25$ ml.

NCERT Example 5 — triangle with angles in ratio $1:3:5$

Angle sum $=180^\circ$, parts sum $=1+3+5=9$. $\angle A=180^\circ\times\dfrac19=20^\circ$, $\angle B=180^\circ\times\dfrac39=60^\circ$, $\angle C=180^\circ\times\dfrac59=100^\circ$. (Check: $20+60+100=180$.)

5. A slice of the pie — pie charts

A pie chart shows each part's share of a whole as a slice of a circle. The full circle is $360^\circ$, and each slice angle is proportional to its value.

$$\text{Slice angle}=\frac{\text{value of that part}}{\text{total of all parts}}\times360^\circ$$

Grades of $40$ students: A $=12$, B $=10$, C $=8$, D $=6$, E $=4$, in ratio $12:10:8:6:4$. Divide by HCF $2$ to simplify: $6:5:4:3:2$ (sum $=20$). Then each angle is $\dfrac{\text{term}}{20}\times360^\circ=\text{term}\times18^\circ$:

  • Grade A $=6\times18^\circ=108^\circ$   Grade B $=5\times18^\circ=90^\circ$
  • Grade C $=4\times18^\circ=72^\circ$   Grade D $=3\times18^\circ=54^\circ$   Grade E $=2\times18^\circ=36^\circ$

(Check: $108+90+72+54+36=360^\circ$.) To draw: draw a circle with radius $AB$, then with a protractor mark each angle one after another from the previous radius, label and colour the slices.

Reading a pie chart — modes of transport

Given slices Bus $120^\circ$, Walk $90^\circ$, Cycle $60^\circ$, Car $60^\circ$, Two-wheeler $30^\circ$. The most common mode is the biggest slice (Bus). Fraction by Car $=\dfrac{60}{360}=\dfrac16$. If $18$ children travel by Car (the $60^\circ$ slice), total surveyed $=18\times\dfrac{360}{60}=108$ children. Walk and Cycle... here Cycle and Car are equal ($60^\circ$ each).

6. Direct proportions (the rule of three)

When two ratios are proportional, $a:b :: c:d$, then $d=\dfrac{bc}{a}$. Such a relationship — where both quantities grow or shrink by the same factor so their quotient stays fixed — is a direct proportion.

$$\frac{x_1}{y_1}=\frac{x_2}{y_2}=k\quad(k\text{ constant})$$
NCERT Example 1 — bricks and workers

If $5$ workers move $4500$ bricks in a day, how many workers move $18000$ bricks? Statement: $4500:18000 :: 5:x$. So $x=\dfrac{18000\times5}{4500}=20$ workers. (More bricks need more workers — same direction.)

Quick test: when one quantity becomes $3$ times, does the other also become $3$ times? If yes, it is direct.

7. Inverse proportions

Sometimes two quantities change by the same factor but in opposite directions — as one increases, the other decreases by the same factor. This is an inverse proportion. Travelling Lucknow→Kanpur: faster speed means less time.

Speed (km/h)5153060
Time (h)18631.5

Notice the product is always $90$ (the fixed distance): $5\times18=15\times6=30\times3=60\times1.5=90$. Speed up $3\times$ ($5\to15$) and time falls to $\tfrac13$ ($18\to6$).

$$xy=k\ \text{(constant)}\quad\Rightarrow\quad x_1y_1=x_2y_2,\qquad \frac{x_1}{x_2}=\frac{y_2}{y_1}$$
NCERT Example 3 — workers laying a road

$20$ workers take $4$ days. How long for $10$ workers? Fewer workers → more days (inverse). $20\times4=10\times y_2\Rightarrow y_2=\dfrac{80}{10}=8$ days.

NCERT Examples 4 & 5 — pumps and provisions

Pumps: $2$ pumps fill a tank in $18$ h. With $4$ pumps: $2\times18=4\times x\Rightarrow x=9$ h. Provisions: food for $80$ students lasts $15$ days; with $20$ more (so $100$ students): $80\times15=100\times x\Rightarrow x=12$ days.

8. Time-and-work (working together)

If a job is "$1$ unit of work," a person who finishes it in $t$ hours does $\dfrac1t$ of it per hour. Add the hourly rates to find the combined rate.

NCERT Example 6 — Ram and Shyam cut vegetables

Ram finishes in $1$ h (rate $1$/h); Shyam in $1.5$ h (rate $\dfrac{1}{1.5}=\dfrac23$/h). Together per hour $=1+\dfrac23=\dfrac53$ units. To do $\dfrac53$ units takes $1$ h, so by direct proportion $\dfrac53:1 :: 1:x$, giving $\dfrac53\times x=1\Rightarrow x=\dfrac{1}{5/3}=\dfrac35$ h. Together they finish in $\dfrac35$ hour.

Shortcut for two workers: if they take $a$ and $b$ hours alone, together they take $\dfrac{ab}{a+b}$ hours. Here $\dfrac{1\times1.5}{1+1.5}=\dfrac{1.5}{2.5}=\dfrac35$ h — same answer.

9. NCERT "Figure it Out" — fully solved

Sharing in a ratio.

  • Cricket practice $3:4:3:5$ of $150$ min. Sum $=15$; one part $=10$ min. Warm-up $=30$, batting $=40$, bowling $=30$, fielding $=50$ minutes.
  • Library $3:2:1$, with $288$ Odiya books. Odiya is the $3$-part $=288$, so $1$ part $=96$. Hindi $=2\times96=192$, English $=1\times96=96$ books.
  • $100$ coins in $4:3:2:1$ ($\textsf{₹}10,\textsf{₹}5,\textsf{₹}2,\textsf{₹}1$). Sum $=10$, one part $=10$ coins. Value $=40\times10+30\times5+20\times2+10\times1=400+150+40+10=\textsf{₹}600$.
  • Triangle sides $3:4:5$: all such triangles are similar but not congruent (same shape, different sizes). Sides $1:3:5$ cannot form a triangle — $1+3=4<5$ breaks the triangle inequality.

Inverse proportion — which pairs? Inverse means product constant.

  • Taps filling a tank & time — inverse; painters & days — inverse; speed of cyclist & time for fixed route — inverse; pages in a book & reading time — direct; cloth length & price at fixed rate — direct; distance a car travels & petrol — direct.
  • Tables: a pair is inverse only if every $x\times y$ is the same. (i) $40\times20=800,\ 25\times32=800$ but $16\times50=800$ — all $800$ → inverse. (ii) $40\times20=800$ but $25\times12.5=312.5$ → not. (iii) $30\times15=450,\ 90\times5=450,\ 150\times3=450,\ 10\times45=450$ → inverse.

Direct / inverse word problems.

  • $24$ pencils cost $\textsf{₹}120$; $20$ pencils? Direct. $\textsf{₹}\dfrac{120}{24}\times20=\textsf{₹}100$.
  • Water for $20$ families lasts $6$ days; $10$ more families ($30$)? Inverse. $20\times6=30\times x\Rightarrow x=4$ days (assume equal daily use per family).
  • $3$ workers paint a fence in $4$ days; with $1$ more ($4$)? Inverse. $3\times4=4\times x\Rightarrow x=3$ days.
  • $6$ h to fill $2$ tanks; time for $5$ tanks? Direct. $\dfrac{6}{2}\times5=15$ h.
  • $25$ rows of $12$ chairs; rows with $20$ chairs? Inverse (total $=300$). $25\times12=20\times x\Rightarrow x=15$ rows.
  • $8$ periods of $45$ min; $9$ periods, same total time? Inverse. $8\times45=9\times x\Rightarrow x=40$ min.
  • Small pump $3$ h, large pump $2$ h, together? $\dfrac{3\times2}{3+2}=\dfrac65=1.2$ h.
  • $42$ machines make toys in $63$ days; machines for $54$ days? Inverse. $42\times63=x\times54\Rightarrow x=49$ machines.
  • Car $2$ h at $60$ km/h; time at $80$ km/h? Inverse. $60\times2=80\times x\Rightarrow x=1.5$ h.

10. Common mistakes to avoid

  • Mixing up the two tests: direct uses equal quotients $\dfrac{x_1}{y_1}=\dfrac{x_2}{y_2}$; inverse uses equal products $x_1y_1=x_2y_2$.
  • For inverse proportion, writing $\dfrac{x_1}{x_2}=\dfrac{y_1}{y_2}$ — it must be flipped: $\dfrac{x_1}{x_2}=\dfrac{y_2}{y_1}$.
  • Forgetting to convert RF units — $60{,}00{,}000$ cm is $60$ km, not $60{,}00{,}000$ m.
  • In ratio sharing, multiplying by the term before finding "one part" $=\dfrac{\text{whole}}{\text{sum of terms}}$.
  • Pie chart: using the raw value as the angle instead of $\dfrac{\text{value}}{\text{total}}\times360^\circ$; slices must add to $360^\circ$.
  • Assuming any ratio of sides forms a triangle — check the triangle inequality (e.g. $1:3:5$ fails).

11. Quick revision checklist

  • Cross-multiply to test proportion: $a\times d=b\times c$.
  • Map RF $1:n$ — multiply map distance by $n$ for ground distance; convert cm→km.
  • Divide $x$ in ratio: one part $=\dfrac{x}{\text{sum of terms}}$, then scale each term.
  • Pie slice $=\dfrac{\text{value}}{\text{total}}\times360^\circ$; simplify the ratio first.
  • Direct: quotient $k$ constant. Inverse: product $k$ constant.
  • Two workers together: $\dfrac{ab}{a+b}$ time.
Practice MCQs
1. Two ratios $a:b$ and $c:d$ are proportional if:
  1. $a+d=b+c$
  2. $a\times d=b\times c$
  3. $a\times b=c\times d$
  4. $a-b=c-d$
Answer: (B) cross-products are equal, $a\times d=b\times c$.
2. On a map with RF $1:60{,}00{,}000$, $1$ cm represents:
  1. $6$ km
  2. $60$ km
  3. $600$ km
  4. $6000$ km
Answer: (B) $60{,}00{,}000$ cm $=60{,}000$ m $=60$ km.
3. Paint is mixed Red : Blue : White $=2:3:5$. In $20$ L of paint, the white paint is:
  1. $4$ L
  2. $6$ L
  3. $10$ L
  4. $12$ L
Answer: (C) White $=20\times\dfrac{5}{10}=10$ L.
4. If $\textsf{₹}900$ is divided between two friends in $4:5$, the smaller share is:
  1. $\textsf{₹}400$
  2. $\textsf{₹}500$
  3. $\textsf{₹}450$
  4. $\textsf{₹}360$
Answer: (A) sum $=9$, one part $=100$; smaller $=4\times100=\textsf{₹}400$.
5. In a pie chart, a part that is $\dfrac14$ of the data takes up an angle of:
  1. $45^\circ$
  2. $60^\circ$
  3. $90^\circ$
  4. $120^\circ$
Answer: (C) $\dfrac14\times360^\circ=90^\circ$.
6. The angle for a value of $8$ out of a total of $40$ in a pie chart is:
  1. $36^\circ$
  2. $54^\circ$
  3. $72^\circ$
  4. $90^\circ$
Answer: (C) $\dfrac{8}{40}\times360^\circ=72^\circ$.
7. If $5$ workers move $4500$ bricks a day, the workers needed for $18000$ bricks are:
  1. $10$
  2. $15$
  3. $20$
  4. $25$
Answer: (C) direct: $\dfrac{18000\times5}{4500}=20$ workers.
8. Two quantities $x$ and $y$ are in inverse proportion when:
  1. $\dfrac{x}{y}=k$
  2. $x+y=k$
  3. $xy=k$
  4. $x-y=k$
Answer: (C) their product $xy$ stays constant.
9. $20$ workers lay a road in $4$ days. The days for $10$ workers (same road) are:
  1. $2$
  2. $6$
  3. $8$
  4. $10$
Answer: (C) inverse: $20\times4=10\times y\Rightarrow y=8$ days.
10. Which table shows inverse proportion? $x:\,40,80$ and $y:$
  1. $y:\,20,40$
  2. $y:\,20,10$
  3. $y:\,80,40$
  4. $y:\,10,20$
Answer: (B) products equal: $40\times20=80\times10=800$.
11. A car takes $2$ h at $60$ km/h. At $80$ km/h it takes:
  1. $1$ h
  2. $1.5$ h
  3. $2$ h
  4. $2.5$ h
Answer: (B) inverse: $60\times2=80\times x\Rightarrow x=1.5$ h.
12. The angles of a triangle are in ratio $1:3:5$. The largest angle is:
  1. $60^\circ$
  2. $90^\circ$
  3. $100^\circ$
  4. $120^\circ$
Answer: (C) sum $=9$ parts; largest $=180^\circ\times\dfrac59=100^\circ$.
Assertion–Reason
A: As the speed of a car increases, the time to cover a fixed route decreases.   R: Speed and time for a fixed distance are in inverse proportion.
Answer: Both A and R are true, and R correctly explains A — speed × time = distance (constant), so they vary inversely.
A: To divide a whole in ratio $a:b:c$, each part equals the whole times its term.   R: Each share equals (whole) $\times \dfrac{\text{its term}}{a+b+c}$.
Answer: A is false (you must divide by the sum of terms first), R is true — R is the correct formula.
Exam-style questions
Q1. The library books are in ratio Odiya : Hindi : English $=3:2:1$. If there are $288$ Odiya books, find the number of Hindi and English books. (3 marks)
Answer: Odiya is $3$ parts $=288$, so $1$ part $=96$. Hindi $=2\times96=192$, English $=1\times96=96$ books.
Q2. Draw the pie-chart angles for grades A : B : C : D : E with $12,10,8,6,4$ students (total $40$). (4 marks)
Answer: Each angle $=\dfrac{\text{students}}{40}\times360^\circ$: A $=108^\circ$, B $=90^\circ$, C $=72^\circ$, D $=54^\circ$, E $=36^\circ$ (sum $=360^\circ$).
Q3. A school has food for $80$ students for $15$ days. If $20$ more students join, for how many days will the food last? (3 marks)
Answer: Inverse proportion. $80\times15=100\times x\Rightarrow x=\dfrac{1200}{100}=12$ days.
Q4. A small pump fills a tank in $3$ hours and a large pump in $2$ hours. How long if both run together? (3 marks)
Answer: Per hour together $=\dfrac13+\dfrac12=\dfrac56$ tank. Time $=\dfrac{1}{5/6}=\dfrac65=1.2$ hours (i.e. $1$ h $12$ min).
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A Square and A Cube · Power Play · A Story of Numbers · Quadrilaterals · Number Play · We Distribute, Yet Things Multiply · Proportional Reasoning – 1 · Fractions in Disguise
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