Chapter 7: Proportional Reasoning-1 (Ratio)

Width: $90÷60=\frac{3}{2}$. Height: $60÷40=\frac{3}{2}$.

Both change by the same factor, 1.5, so D is an enlarged copy of A and looks similar.

Both are multiplied by 3/2 (1.5); the factors are the same, so A and D look similar.

$90÷60=40\times \frac{3}{2}÷40=\frac{3}{2}$. Check: $60\times \frac{3}{2}:40\times \frac{3}{2}=90:60$ ✓

Both ratios reduce to $3:2$.

By 3/2.

All three cups are 50 mL. What changes is how much of the cup is decoction:

• Regular, 15 : 35 = 3 : 7: decoction is $\frac{15}{50}=30\%$ of the cup.
• 20 : 30 = 2 : 3: $\frac{20}{50}=40\%$, more decoction for the same milk, so stronger.
• 10 : 40 = 1 : 4: only $20\%$, so lighter.

For the table, compare each mix with the regular 3 : 7 (decoction 30% of the total):

Strong, light, strong, regular, light. A mix is stronger when it has more decoction per mL of milk than 3 : 7.

Reduce both ratios, or check cross products ($ad=bc$):

1. $12:21=4:7$. True.
2. $24:6=4:1$, not $8:3$. False.
3. $7\times 7=49\ne 144=12\times 12$. False.
4. $21:6=7:2$ and $35:10=7:2$. True.
5. $12:18=2:3$ but $28:12=7:3$. False.
6. $24:8=3:1$ and $9:3=3:1$. True.

(i), (iv) and (vi) are true.

Multiply both terms by the same number: ×2, ×3 and ×10 give the three ratios below.

8 : 18, 12 : 27, 40 : 90 (any 4k : 9k works).

$18:24=3:4$, so the second term is always $\frac{4}{3}$ of the first:

• $3:4$
• $12:16$
• $20:\frac{80}{3}$, i.e. $20:26\frac{2}{3}$
• $27:36$

3 : 4, 12 : 16, 20 : 26⅔ (= 80/3), 27 : 36

Measure the two sides of each rectangle (for the tilted ones, measure along their edges). Sizes on the printed page are approximately:

A and D are similar (both 1 : 3). C and E are similar (both 2 : 5). B (2 : 3) is not similar to any of the others.

Turning a rectangle does not change its shape.

A and D are similar (1 : 3); C and E are similar (2 : 5); B is not similar to any other.

The given rectangle measures about 3.5 cm × 2.1 cm, so width : height ≈ 5 : 3. Draw:

• Smaller: 2.5 cm × 1.5 cm (factor $\frac{5}{7}$)
• Bigger: 5 cm × 3 cm, or 7 cm × 4.2 cm (factor 2)

Classmates' rectangles may have different sizes, because each person may choose a different factor. They are not wrong as long as width : height is still 5 : 3. All such rectangles are similar: the same shape at different sizes.

Yes, e.g. 2.5 cm × 1.5 cm and 5 cm × 3 cm (ratio 5 : 3). Different sizes are fine as long as the ratio is the same.

(a) One repeat is 5 bricks wide in each of the 3 rows:

• Coloured: $3+2+1=6$
• Grey: $2+3+4=9$

Grey : coloured $=9:6=3:2$.

(b) One repeat is 4 bricks wide in each of the 7 rows. Coloured bricks per row (top to bottom): 1, 2, 2, 2, 2, 2, 1.

• Coloured: 12
• Grey: $28-12=16$

Grey : coloured $=16:12=4:3$.

(a) 3 : 2 (b) 4 : 3

This is an activity, so your numbers will differ. A sample set of measurements for a 13-year-old:

• head 22 cm, torso 55 cm, arm 66 cm, leg 77 cm
• head : torso = 22 : 55 = 2 : 5
• torso : arms = 55 : 66 = 5 : 6
• torso : legs = 55 : 77 = 5 : 7

To draw, choose a unit, say 1 cm for every 11 cm of body. Then draw the head 2 cm, the torso 5 cm, the arms 6 cm and the legs 7 cm.

A drawing with these ratios looks realistic: it is the body at a smaller scale. If one part is changed by a different factor (say, very long arms), the figure looks distorted, like image B of the tiger.

Activity: keep all four lengths in the measured ratios (e.g. 2 : 5, 5 : 6, 5 : 7). Proportional drawings look realistic; non-proportional ones look distorted.

First change 4 hours to 240 minutes, so both times use the same unit.

Cross multiplication: From $150:90::240:x$, we get $x=\frac{240\times 90}{150}=144$.

Unit time: 150 min → 90 km, so 30 min → 18 km. Since 240 min = 8 × 30 min, the distance is $8\times 18=144$ km.

Speed: $90\text{ km}÷2.5\text{ h}=36$ km/h, so in 4 h it goes $36\times 4=144$ km.

144 km.

Compare prices for the same weight:

• Himachal: 200 g is $\frac{1}{5}$ kg, so 1 kg costs $5\times$ ₹200 = ₹1,000.
• Meghalaya: 1 kg costs ₹800.

Himachal tea costs more per kg. It comes from a small farmer who produces small amounts, often by hand, while a large estate can produce and sell in bulk.

Himachal tea: ₹1,000 per kg against ₹800 per kg.

Every ingredient is multiplied by the same factor. For example, if a recipe of poha serves 4, the factor for 15 guests is $\frac{15}{4}=3.75$:

Multiply every ingredient by guests ÷ people the recipe serves (here 15/4 = 3.75).

A year has about 52 weeks:

$$\frac{940,000,000}{52}\approx 18,076,923\text{ km}$$

Using 365 days instead gives $940,000,000\times \frac{7}{365}\approx 18,027,397$ km.

About 18 million km (≈ 1.8 crore km) per week.

Total length of walls:

• Top: $9+15=24$ ft
• Two side walls: $12+12=24$ ft
• Inner wall: 12 ft
• Bottom of the two rooms: $9+15=24$ ft
• Small room: $9+9+6=24$ ft

Total $=24+24+12+24+24=108$ ft.

From $10:1450::108:x$, we get $x=\frac{1450\times 108}{10}=15,660$.

About 15,660 bricks (for 108 ft of wall).

No. A higher speed means less time, so time is not proportional to speed.

The distance stays the same: $50\times 2=100$ km. At 75 km/h, the time is $\frac{100}{75}=\frac{4}{3}$ h $=$ 1 h 20 min.

(Writing 50 : 2 :: 75 : 3 would wrongly give 3 hours.)

It takes less time: 1 h 20 min. The Rule of Three does not apply, because time decreases as speed increases.

$6:180=1:30$ but $2:154=1:77$, so these are not proportional. Price per mL:

Prices also cover packing, transport and marketing, and they are set to suit different buyers. That is why they do not grow in proportion to the volume.

• Sachets: low cost per pack, which suits people who buy small amounts. But they create a great deal of plastic waste that is hard to recycle.
• Large bottles: much less packaging per mL.

For the ecological footprint, the company could sell refill packs and big recyclable bottles. Customers could buy the large size and reuse or refill the bottles.

Not proportional (1 : 30 vs 1 : 77); price per mL varies (₹0.33 to ₹0.86) because of packing and pricing choices. Large or refill packs reduce plastic waste.

• Equally: 6 each, ratio $6:6=1:1$.
• Partner gets 5: you get 7, ratio $5:7$.
• 12 in 3 : 1: $3+1=4$ groups of $12÷4=3$, giving 9 and 3.
• 42 in 4 : 3: $4+3=7$ groups of $42÷7=6$, giving 24 and 18.

1 : 1; 5 : 7; 9 and 3; 24 and 18.

Divide into $2+3=5$ parts of ₹4,500 ÷ 5 = ₹900 each:

• First part: $2\times 900=$ ₹1,800
• Second part: $3\times 900=$ ₹2,700

₹1,800 and ₹2,700

$1+5=6$ parts, and $240÷6=40$ mL per part:

• Acid: 40 mL
• Water: $5\times 40=200$ mL

Acid 40 mL, water 200 mL.

$3+5=8$ parts, and $40÷8=5$ mL per part:

• Blue: $3\times 5=15$ mL
• Yellow: $5\times 5=25$ mL

After adding 20 mL of yellow, there is $25+20=45$ mL of yellow. So blue : yellow $=15:45=1:3$.

15 mL blue and 25 mL yellow; new ratio 1 : 3.

$2+1=3$ parts, and $6÷3=2$ cups per part:

• Rice: 4 cups
• Urad dal: 2 cups

4 cups rice and 2 cups urad dal.

Take one bucket as 8 equal parts: 3 parts red and 5 parts yellow.

The added bucket of the same size is 8 more parts of yellow, so yellow becomes $5+8=13$ parts.

Red : yellow $=3:13$.

3 : 13 (taking both buckets to be the same size).

$$F=\frac{9}{5}\times 25+32=45+32=77$$

Back again: $C=\frac{5}{9}(77-32)=\frac{5}{9}\times 45=25$ ✓

25 °C = 77 °F.

$600:900$. Divide both terms by the HCF, 300.

2 : 3

One bus holds $162÷3=54$ people. Then $204÷54=3.78$, so 3 buses are not enough and we need 4 buses.

They have $4\times 54=216$ seats. That leaves $216-204=12$ seats empty, so the buses will not all be full.

4 buses; not all full (12 seats will be empty).

Compare the people per sq km (population density):

• Delhi: $30,000,000÷1484\approx 20,216$
• Mumbai: $20,000,000÷550\approx 36,364$

Mumbai has fewer people in total, but far more on each sq km.

Mumbai (about 36,000 people per sq km against about 20,000 in Delhi).

The neck is $\frac{4}{4+6}=\frac{4}{10}$ of the height.

• The crane: $0.4\times 155=62$ cm.
• If your height is 150 cm: neck $=0.4\times 150=60$ cm.

Neck = 4/10 of your height (e.g. 60 cm if you are 150 cm tall).

From $2\frac{1}{2}:\frac{3}{7}::x:9$:

$$x=\frac{\frac{5}{2}\times 9}{\frac{3}{7}}=\frac{45}{2}\times \frac{7}{3}=\frac{105}{2}=52\frac{1}{2}$$

52½ palas.

After $x$ years, $(1+x):(5+x)=1:2$. So $2(1+x)=5+x$, which gives $x=3$.

Their ages will then be 4 and 8, and $4:8=1:2$ ✓

When Harmain is 4 years old (her brother will be 8).

From $37:2::x:1$, we get $x=\frac{37}{2}=18.5$.

18.5 kg.

The area is $200\times 500=1,00,000$ sq ft. Since 1 acre $=43,560$ sq ft:

$$\text{Manure}=10\times \frac{100000}{43560}\approx 22.96\text{ tonnes}$$

About 23 tonnes (≈ 22.96 t) of cow manure.

10 L = 10,000 mL = 20 mugs, so the time is $20\times 15=300$ s.

300 seconds = 5 minutes.

$$\text{Cost}=15,00,000\times \frac{2400}{43560}\approx 82,645$$

About ₹82,645.

• Oxen: $20\times 6=120$ hours.
• Tractor: 4 times faster means $\frac{1}{4}$ of the time, so $120÷4=30$ hours (1.5 hours per acre).

Oxen: 120 hours; tractor: 30 hours.

Copper:

• Mass $=\frac{3}{4}\times 7.74=5.805$ g
• Cost $=5.805\times \frac{906}{1000}\approx$ ₹5.26

Nickel:

• Mass $=\frac{1}{4}\times 7.74=1.935$ g
• Cost $=1.935\times \frac{1341}{1000}\approx$ ₹2.59

Total ≈ ₹5.26 + ₹2.59 = ₹7.85.

Copper ≈ ₹5.26 + nickel ≈ ₹2.59, so about ₹7.85 of metal in a ₹10 coin.

Here — is a horizontal line and | is a vertical line. Shaded cells were given in the puzzle.

How to start:

1. Wherever two equal symbols sit side by side, the cells on both sides of the pair must hold the other symbol.
2. Wherever two equal symbols have a gap between them, the gap must hold the other symbol.
3. Once a row or column has 3 of one symbol, fill the rest with the other.

Each of the three puzzles then has exactly one solution.

Puzzle 1

Puzzle 2

Puzzle 3

Each puzzle has a unique solution, shown in the grids above (every row and column has three of each symbol, no three in a line, all rows and columns different).