Proportional Reasoning-1 Class 8: Ratios, Proportion, Rule of Three and Unit Conversions

What Proportional Reasoning-1 is about

Key concepts

Proportional change: multiply, don't add

NCERT section 7.1 Observing Similarity in Change

Two images look similar (same shape, different size) when their width and height change by the same factor.

• 80 mm by 50 mm and 40 mm by 25 mm: both sides halved, so they look alike.
• 80 mm by 50 mm and 60 mm by 30 mm: the width shrank by 20 mm and the height by 20 mm too, but the factors are $\frac{60}{80}=\frac{3}{4}$ and $\frac{30}{50}=\frac{3}{5}$, which differ. The second picture looks squashed.

So adding or subtracting the same amount does not keep things proportional; multiplying by the same factor does.

This also explains ages: a girl of 4 and her father of 32 are in the ratio $1:8$, but 10 years later (14 and 42) the ratio is $1:3$. Adding the same number of years changes the ratio.

Worked example

Picture P is 80 mm wide and 50 mm high. Picture Q is 40 mm by 25 mm and picture R is 60 mm by 30 mm. Which picture looks like a resized P, and why?

Try it first, then show the solutionSolution

1. P to Q: width factor $\frac{40}{80}=\frac{1}{2}$, height factor $\frac{25}{50}=\frac{1}{2}$: the same.
2. P to R: width factor $\frac{60}{80}=\frac{3}{4}$, height factor $\frac{30}{50}=\frac{3}{5}$: different.
3. So Q is a proportional resize of P; R is distorted.

Answer: Q, because both sides were multiplied by the same factor $\frac{1}{2}$.

Ratios and their simplest form

NCERT section 7.3 Ratios in their Simplest Form

A ratio $a:b$ compares two quantities: for every $a$ units of the first there are $b$ units of the second. $a$ and $b$ are the terms of the ratio.

• Both terms must be in the same unit before writing a ratio (convert 1 kg to 1000 g, 4 hours to 240 minutes).
• To write a ratio in its simplest form, divide both terms by their HCF: $84:126$; HCF is 42; simplest form $2:3$.
• Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio: $4:9=8:18=12:27$.
• Order matters: width to height $3:2$ is not the same as height to width $2:3$.

Worked example

Write $84:126$ and $1.5\text{ kg}:600\text{ g}$ in their simplest forms.

Try it first, then show the solutionSolution

1. HCF(84, 126) = 42, so $84:126=2:3$.
2. Same units: $1.5$ kg $=1500$ g, so the ratio is $1500:600$.
3. HCF(1500, 600) = 300, so $1500:600=5:2$.

Answer: $2:3$ and $5:2$

Proportion and cross multiplication

NCERT section 7.3 Ratios in their Simplest Form

Two ratios are in proportion (written $a:b::c:d$) when their simplest forms are the same, that is, when both terms of one are the same multiple $f$ of the terms of the other: $c=fa$ and $d=fb$.

From $f=\frac{c}{a}=\frac{d}{b}$ we get the cross multiplication rule:

**$a:b::c:d$ exactly when $ad=bc$.**

So $12:18::30:45$ because $12\times 45=540=18\times 30$.

Finding a missing term: if $15:35::x:63$, then $35x=15\times 63$, so $x=27$. Alternatively spot the factor: $63=35\times \frac{9}{5}$, so $x=15\times \frac{9}{5}=27$.

Worked example

Are $12:18$ and $30:45$ proportional? Find $x$ if $15:35::x:63$.

Try it first, then show the solutionSolution

1. $12\times 45=540$ and $18\times 30=540$: equal, so they are proportional (both are $2:3$).
2. For the missing term: $15\times 63=35\times x$.
3. $x=\frac{945}{35}=27$.

Answer: Yes, proportional; $x=27$.

The Rule of Three (trairasika)

NCERT section 7.4 Problem Solving with Proportional Reasoning

Many problems link four quantities in proportion, with three known and one unknown. Ancient Indian mathematicians, including Aryabhata, called these Rule of Three problems:

• *pramana* (measure, $a$), *phala* (fruit, $b$), *ichchha* (requisition, $c$), and the unknown *ichchhaphala* (yield, $d$);
• *pramana* : *phala* :: *ichchha* : *ichchhaphala*, so $d=\frac{b\times c}{a}$: multiply the fruit by the requisition and divide by the measure.

Steps:
1. Check that the quantities really are proportional (double one, does the other double?).
2. Put both ratios in the same units.
3. Write $a:b::c:d$ and cross-multiply.

Example: a car covers 45 km in 50 minutes. In 3 hours (180 minutes): $50:45::180:d$, so $d=\frac{45\times 180}{50}=162$ km. Writing $50:45::3:d$ would be wrong, because minutes and hours are mixed.

Worked example

8 kg of rice costs ₹360. Find the cost of 15 kg using the Rule of Three, then check with the price of 1 kg.

Try it first, then show the solutionSolution

1. Weight and cost are proportional: $8:360::15:d$.
2. $8d=360\times 15=5400$, so $d=675$.
3. Check: 1 kg costs $360÷8=₹45$, and $45\times 15=₹675$.

Answer: ₹675

Comparing rates and spotting non-proportional situations

NCERT section 7.4 Problem Solving with Proportional Reasoning

Which is the better buy? Compare the price of the same amount. 750 g for ₹180 means ₹240 per kg; 2 kg for ₹460 means ₹230 per kg, so the 2 kg pack is cheaper per kilogram. The weight-to-price ratios ($750:180=25:6$ and $2000:460=100:23$) are not proportional.

Bigger packs and prices: in real shops, price is often not proportional to quantity (a large bottle may cost less per mL than a sachet), so always compute the rate before assuming proportionality.

Not every 'three numbers' problem is a Rule of Three. If a journey takes 3 hours at 60 km/h, going at 90 km/h does not take longer in proportion; it takes less time. The distance (180 km) stays fixed, so time $=180÷90=2$ hours. When one quantity goes up as the other goes down, writing $60:3::90:?$ is wrong. (Such inverse proportions come in Proportional Reasoning-2.)

Worked example

Pack A has 750 g of dal for ₹180; pack B has 2 kg for ₹460. Which is cheaper per kilogram, and by how much?

Try it first, then show the solutionSolution

1. A: $180÷0.75=₹240$ per kg.
2. B: $460÷2=₹230$ per kg.
3. B is cheaper by $240-230=₹10$ per kg.

Answer: Pack B, by ₹10 per kg.

Sharing in a ratio

NCERT section 7.5 Sharing, but Not Equally!

To divide a quantity $x$ in the ratio $m:n$:
1. Count the total number of equal groups: $m+n$.
2. One group is $\frac{x}{m+n}$.
3. The parts are $m\times \frac{x}{m+n}$ and $n\times \frac{x}{m+n}$.

Check: the parts add to $x$ and are in the ratio $m:n$.

Example: ₹6,300 in the ratio $4:5$: $4+5=9$ groups of $6300÷9=700$, so ₹2,800 and ₹3,500.

Profits are often shared in the ratio of the money invested: investments of ₹75,000 and ₹25,000 are in the ratio $3:1$, so a profit of ₹8,000 is shared as ₹6,000 and ₹2,000.

Worked example

Divide ₹6,300 between two sisters in the ratio $4:5$.

Try it first, then show the solutionSolution

1. Total groups $=4+5=9$.
2. One group $=6300÷9=₹700$.
3. Shares: $4\times 700=₹2800$ and $5\times 700=₹3500$; check $2800+3500=6300$.

Answer: ₹2,800 and ₹3,500

Changing the ratio of a mixture

NCERT section 7.5 Sharing, but Not Equally!

In mixture problems one ingredient is added while the other stays the same. Work in three steps:
1. Find the amount of each ingredient now (share the total in the given ratio).
2. Keep the unchanged ingredient fixed and use the new ratio to find how much of the other is needed.
3. Subtract to find how much must be added.

Example: 60 kg of sand and cement in the ratio $4:1$ has 48 kg sand and 12 kg cement. For a ratio of $3:1$, cement must be $\frac{48}{3}=16$ kg, so add 4 kg of cement. (For $5:1$ instead, sand must be $5\times 12=60$ kg, so add 12 kg of sand.)

Adding one ingredient changes the ratio; adding both in the original ratio keeps it the same.

Worked example

A 5 kg brass block has copper and zinc in the ratio $7:3$. How much zinc must be melted in to change the ratio to $7:4$?

Try it first, then show the solutionSolution

1. Copper $=\frac{7}{10}\times 5=3.5$ kg; zinc $=1.5$ kg.
2. Copper stays 3.5 kg. For $7:4$, zinc $=\frac{4}{7}\times 3.5=2$ kg.
3. Zinc to add $=2-1.5=0.5$ kg.

Answer: 0.5 kg (500 g) of zinc.

Unit conversions

NCERT section 7.6 Unit Conversions

Proportion problems often need units converted first. Useful conversions from the book:

• Length: 1 metre $=3.281$ feet.
• Area: 1 square metre $=10.764$ square feet; 1 acre $=43,560$ square feet; 1 hectare $=10,000$ square metres $=2.471$ acres.
• Volume: 1 mL $=1$ cubic centimetre (cc); 1 litre $=1000$ mL.
• Temperature: $F=\frac{9}{5}C+32$ and $C=\frac{5}{9}(F-32)$; $0°C=32°F$.

Length and area conversions are proportional (multiply by a fixed factor). Temperature conversion is not simply proportional because of the '+ 32': doubling °C does not double °F.

Example: 35 °C $=\frac{9}{5}\times 35+32=63+32=95$ °F.

Worked example

A plot measures 300 ft by 363 ft. Find its area in acres. If 8 tonnes of manure are needed per acre, how much is needed?

Try it first, then show the solutionSolution

1. Area $=300\times 363=1,08,900$ sq ft.
2. In acres: $108900÷43560=2.5$ acres.
3. Manure: $8\times 2.5=20$ tonnes.

Answer: 2.5 acres; 20 tonnes of manure.

Maths in real life: Proportional Reasoning-1

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Ratios and Proportion Word Problems
• Sharing in a Ratio, Mixtures and Unit Conversions

Proportional Reasoning-1 formula sheet

Simplest form

$a:b=\frac{a}{h}:\frac{b}{h},h=\text{HCF}(a,b)$

Proportion

$a:b::c:d⟺ad=bc$

Rule of Three

$d=\frac{b\times c}{a}$

pramana : phala :: ichchha : ichchhaphala

Sharing in a ratio

$x\text{ in }m:n\to \frac{mx}{m+n},\frac{nx}{m+n}$

Unit rate

$\text{rate}=\frac{\text{price}}{\text{quantity}}$

Length

$1\text{ m}=3.281\text{ ft}$

Area

$1\text{ acre}=43560\text{ sq ft},1\text{ ha}=10000{\text{ m}}^2=2.471\text{ acres}$

Volume

$1\text{ mL}=1\text{ cc},1\text{ L}=1000\text{ mL}$

Temperature

$F=\frac{9}{5}C+32,C=\frac{5}{9}(F-32)$

How the school exam asks Proportional Reasoning-1

The question patterns to expect, and the method that works for each.

Tricks to remember Proportional Reasoning-1

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Same units first

  'Minutes with minutes, grams with grams' before any ratio is written.

  Use it when: Avoiding the most common setup error.

• Cross the means and extremes

  In $a:b::c:d$, the outer pair times equals the inner pair times: $ad=bc$.

  Use it when: Testing proportions and finding missing terms.

• Fruit times wish over measure

  Rule of Three: phala × ichchha ÷ pramana.

  Use it when: Remembering Aryabhata's rule.

• Share by groups

  Ratio $m:n$ means $m+n$ equal groups; find one group, then multiply.

  Use it when: Dividing in a ratio.

• Freezing at 32

  Water freezes at 0 °C = 32 °F; multiply by 9/5, then add 32.

  Use it when: Celsius to Fahrenheit.

Proportional Reasoning-1: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Proportional Reasoning-1

The slips examiners see most often, and what to write instead.

• ✗Mistake: Mixing units inside a proportion

  Numbers are copied straight from the question.

  Wrong working

  45 km in 50 min, so in 3 hours: $50:45::3:d$, $d=2.7$ km.

  Right working

  3 hours = 180 min: $50:45::180:d$, so $d=162$ km.

  ✓Do this: Convert to one unit before writing the proportion.

  Usually costs: The whole answer.

• ✗Mistake: Using the Rule of Three for an inverse situation

  Every three-number problem is treated as direct proportion.

  Wrong working

  3 hours at 60 km/h, so at 90 km/h: $60:3::90:t$, $t=4.5$ hours.

  Right working

  Faster means less time. Distance $=180$ km, so time $=180÷90=2$ hours.

  ✓Do this: Ask: when one quantity goes up, does the other go up too?

  Usually costs: Method and answer marks.

• ✗Mistake: Adding the same number to keep a ratio

  Additive thinking replaces multiplicative thinking.

  Wrong working

  Recipe for 4 uses 300 g rice and 150 g dal; for 10 people use 306 g and 156 g.

  Right working

  Multiply by $\frac{10}{4}=2.5$: 750 g rice and 375 g dal.

  ✓Do this: Scale every quantity by the same factor.

  Usually costs: The whole answer.

• ✗Mistake: Dividing by the wrong total when sharing

  The amount is divided by one term instead of the sum of the terms.

  Wrong working

  ₹6,300 in $4:5$: $6300÷4$ and $6300÷5$.

  Right working

  $6300÷9=700$; shares ₹2,800 and ₹3,500.

  ✓Do this: Always divide by $m+n$ first.

  Usually costs: 1-2 marks.

• ✗Mistake: Changing both ingredients in a mixture problem

  The new ratio is applied to the new total directly.

  Wrong working

  60 kg at $4:1$ to become $3:1$: cement $=\frac{1}{4}\times 60=15$ kg, so add 3 kg.

  Right working

  Sand stays 48 kg; cement must be $48÷3=16$ kg; add 4 kg (new total 64 kg).

  ✓Do this: Keep the ingredient that is not added fixed.

  Usually costs: Answer mark.

NCERT exercise guide: Proportional Reasoning-1

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Proportional Reasoning-1

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which ratio is proportional to $5:8$?
   (a) $10:20$
   (b) $25:40$
   (c) $15:16$
   (d) $8:5$

   Hint

   Multiply both terms by the same number.

   Answer

   (b) $25:40$

2. Q2 · 1 mark · MCQExam level

   1 hectare is equal to
   (a) 1,000 m²
   (b) 10,000 m²
   (c) 43,560 m²
   (d) 2.471 m²

   Hint

   Unit conversions section.

   Answer

   (b) 10,000 m²

3. Q3 · 2 marks · Very short answerExam level

   Divide ₹7,200 in the ratio $5:7$.

   Hint

   12 groups.

   Answer

   ₹3,000 and ₹4,200

4. Q4 · 1 mark · Very short answerExam level

   Convert 45 °C to °F.

   Hint

   $F=\frac{9}{5}C+32$.

   Answer

   $81+32=113$ °F

5. Q5 · 2 marks · Very short answerHarder exam level

   A drink is made with concentrate and water in the ratio $2:7$. How much of each is needed for 36 cups of drink?

   Hint

   9 groups of 4 cups.

   Answer

   8 cups of concentrate and 28 cups of water.

6. Q6 · 3 marks · Short answerHarder exam level

   Last year 4 identical buses carried 216 people and were all full. This year 300 people are going. How many buses are needed, and will they all be full?

   Hint

   Find people per bus.

   Answer

   Each bus holds $216÷4=54$. $300÷54\approx 5.6$, so 6 buses; 5 are full (270) and the sixth carries 30, so not all full.

7. Q7 · 3 marks · Short answerHarder exam level

   A trip takes 3 hours at 60 km/h. A student writes $60:3::90:t$ to find the time at 90 km/h. Explain the error and find the correct time.

   Hint

   Faster means less time.

   Answer

   Speed and time are not directly proportional here; faster speed means less time. Distance $=60\times 3=180$ km, so time $=180÷90=2$ hours.

8. Q8 · 3 marks · Short answerHarder exam level

   Blue and yellow paint are mixed in the ratio $4:5$ to make 72 mL of green. How much of each is used? If 18 mL of yellow is added, what is the new ratio?

   Hint

   Keep blue fixed.

   Answer

   Blue 32 mL, yellow 40 mL; new ratio $32:58=16:29$.

9. Q9 · 2 marks · Very short answerHarder exam level

   One acre of land costs ₹21,78,000. Find the cost of a 2,000 sq ft plot at the same rate.

   Hint

   Price per sq ft.

   Answer

   $2178000÷43560=₹50$ per sq ft; $2000\times 50=₹1,00,000$.

10. Q10 · 5 marks · Long answerHarder exam level

    A recipe for 4 people uses 300 g rice, 150 g dal and 450 mL water. (a) Find the quantities for 10 people. (b) Write rice : dal : water for one person. (c) Show that rice : dal stays $2:1$ after scaling. (d) A student instead adds 6 people's worth by adding 6 g to each. Why is that wrong?

    Hint

    Scale factor $\frac{10}{4}$.

    Answer

    (a) $\times 2.5$: 750 g rice, 375 g dal, 1,125 mL water. (b) 75 g, 37.5 g, 112.5 mL. (c) $750:375=2:1$. (d) Proportional change is multiplicative; adding the same amount changes the ratios and gives far too little food.

11. Q11 · 3 marks · Short answerHOTS / competency

    A sister is 2 and her brother is 8. When will their ages be in the ratio $1:2$? Will the ratio ever become $1:1$?

    Hint

    Add $t$ to both ages.

    Answer

    $2(2+t)=8+t$ gives $t=4$ (ages 6 and 12). The difference is always 6 years, so the ages are never equal: the ratio never becomes $1:1$.

12. Q12 · 3 marks · Short answerHOTS / competency

    A 6 kg bronze block has copper and tin in the ratio $5:1$. How much tin must be added to make the ratio $4:1$? What is the new mass?

    Hint

    Copper stays the same.

    Answer

    Copper 5 kg, tin 1 kg. Tin needed $=\frac{1}{4}\times 5=1.25$ kg, so add 0.25 kg. New mass 6.25 kg.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.When do two pictures look similar?Show answer

   When their width and height are multiplied by the same factor.

2. 2.What does the ratio $a:b$ mean?Show answer

   For every $a$ units of the first quantity there are $b$ units of the second.

3. 3.How do you write a ratio in its simplest form?Show answer

   Divide both terms by their HCF.

4. 4.What does $a:b::c:d$ mean?Show answer

   The two ratios are in proportion (same simplest form).

5. 5.State the cross multiplication rule.Show answer

   $a:b::c:d$ exactly when $ad=bc$.

6. 6.Name the three given quantities in the Rule of Three.Show answer

   Pramana (measure), phala (fruit) and ichchha (requisition).

7. 7.How do you divide $x$ in the ratio $m:n$?Show answer

   Parts are $\frac{mx}{m+n}$ and $\frac{nx}{m+n}$.

8. 8.How many square feet are in an acre?Show answer

   43,560.

9. 9.How many acres are in a hectare?Show answer

   About 2.471.

10. 10.Convert 0 °C to Fahrenheit.Show answer

    32 °F.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you tell whether two rectangles are similar from their sides?
• ☐Can you simplify a ratio with different units?
• ☐Can you test a proportion by cross multiplication?
• ☐Can you fill missing terms in proportional ratios?
• ☐Can you solve a Rule of Three problem with unit conversion?
• ☐Can you decide which pack is the better buy?
• ☐Can you explain why speed and time are not in direct proportion?
• ☐Can you divide an amount in a ratio $m:n$?
• ☐Can you work out how much to add to change a mixture's ratio?
• ☐Can you convert acres, hectares, square feet and °C to °F?

Proportional Reasoning-1: frequently asked questions