Chapter 3: Proportional Reasoning-2 (Ratio)

1 m = 100 cm and 1 km = 1000 m, so 1 km $=1,00,000$ cm.

$$60,00,000÷1,00,000=60$$

60 km, so on a 1 : 60,00,000 map, 1 cm stands for 60 km.

On a true 1 : 60,00,000 map, 1 cm = 60 km. Measure in a straight line between the city dots:

• Bengaluru – Chennai: about 4.8 cm, so $4.8\times 60\approx$ 290 km.
• Mangaluru – Chennai: about 9.8 cm, so $9.8\times 60\approx$ 590 km.

The map printed in the textbook is marked "not to scale", so a ruler on that page will not give exactly these lengths. With an atlas map, use its own RF the same way.

Different maps have different scales, but each converts back to roughly the same ground distance. Small differences come from measuring errors and from how thick the dots are.

About 290 km (Bengaluru–Chennai) and about 590 km (Mangaluru–Chennai); maps with other scales give nearly the same distances.

5 parts = 10 L, so 1 part = 2 L. Red = 4 L and blue = 6 L.

Total $=4+6+10=20$ L.

20 litres.

$3+4+3+5=15$ parts, and $150÷15=10$ minutes per part:

• Warm-up/cool-down: 30 min
• Batting: 40 min
• Bowling: 30 min
• Fielding: 50 min

30 min, 40 min, 30 min and 50 min.

3 parts = 288, so 1 part = 96:

• Hindi: $2\times 96=192$
• English: $1\times 96=96$

192 Hindi books and 96 English books.

$4+3+2+1=10$ parts of 10 coins each:

• 40 coins of ₹10 = ₹400
• 30 coins of ₹5 = ₹150
• 20 coins of ₹2 = ₹40
• 10 coins of ₹1 = ₹10

₹600

Choose any unit, for example 3 cm, 4 cm and 5 cm, or 6 cm, 8 cm and 10 cm. Construct by SSS:

1. Draw the base.
2. Draw arcs with the other two lengths from its ends.
3. Join the meeting point of the arcs to both ends.

Not congruent: they have the same shape (in fact each has a right angle, since $3^2+4^2=5^2$) but different sizes. They are similar, not congruent. Congruent triangles need equal sides, not just equal ratios.

Draw e.g. 3 cm, 4 cm, 5 cm. All such triangles have the same shape (and a right angle) but need not be congruent, because the sizes can differ.

No. With sides $k,3k,5k$, the two shorter sides add up to $4k$, which is less than $5k$.

In any triangle, the sum of two sides must be greater than the third side. The two short sides would not reach each other.

No: 1 + 3 < 5, so the triangle inequality fails.

Winter $=360-90-120=150$. With 360 people, each person = 1°:

• Summer 90°
• Rainy 120°
• Winter 150°

Summer 90°, Rainy 120°, Winter 150°.

1% of 360° = 3.6°:

• Entertainment: 50% → 180°
• Sports: 25% → 90°
• News: 15% → 54°
• Information: 10% → 36°

180°, 90°, 54°, 36°.

Collect the numbers, then use angle $=\frac{\text{number}}{\text{total}}\times 360°$. A sample class of 40 students:

Use angle = (students ÷ total) × 360°; check that the angles add to 360°.

No. That proportion would give $x=6$ hours, but a faster vehicle takes less time. Doubling the speed halves the time, so the trip takes 1.5 hours.

The table changes by the same factor in opposite directions:

• Walk → bicycle: speed ×3, time ÷3 (18 → 6)
• Bicycle → motorcycle: ×2 and ÷2 (6 → 3)
• Motorcycle → car: ×2 and ÷2 (3 → 1.5)
• Walk → car: ×12 and ÷12 (18 → 1.5)

In every case, speed × time = 90 km.

No: time decreases as speed increases (1.5 h by car). Speed × time = 90 for every mode, so this is an inverse proportion.

Check whether $xy$ is constant:

1. 800, 800, 800, 800, so inverse.
2. 800, 800, 312.5, 128, so not inverse. (Here $x:y=2:1$ throughout, which is a direct proportion.)
3. 450, 450, 450, 450, so inverse.

(i) and (iii) are in inverse proportion; (ii) is not (it is direct).

$k=16\times 9=144$:

• $x=12$: $y=144÷12=12$
• $y=48$: $x=144÷48=3$
• $x=36$: $y=144÷36=4$

x: 16, 12, 3, 36; y: 9, 12, 48, 4.

Directly: at a fixed rate of working, twice the work takes twice the time.

What is inversely proportional is the time and the number of people (or the rate of working).

Directly proportional (for a fixed rate of working).

• Inverse: (i), (ii) and (iv). More of one means proportionally less of the other.
• Direct: (iii), (v) and (vi). Both go up together by the same factor.

(i), (ii) and (iv) are inverse; (iii), (v) and (vi) are direct.

This is a direct proportion. One pencil costs $120÷24=$ ₹5, so 20 pencils cost ₹100.

₹100

This is an inverse proportion: $20\times 6=30\times x$, so $x=4$ days.

Assumptions:

• Every family uses the same amount of water each day.
• Daily use does not change.
• The tank is not refilled and does not lose water in other ways.

4 days, assuming equal and steady use by every family and no refilling.

In each ring the darker (blue) part is the sleep time, as a fraction of 24 hours. Order the rings by how much is blue and match them with the hours in increasing order:

Giraffe 2.5, elephant 3.5, human 8, dog 10.5, cat 13, squirrel 15, python 18, bat 20 hours.

The car slice is $360°-(90°+120°+60°+60°)=30°$.

1. Bus (120°, the largest slice).
2. Car: $\frac{30}{360}=\frac{1}{12}$.
3. $\frac{1}{12}$ of the total is 18, so $18\times 12=$ 216 children were surveyed. There is no taxi slice, so 0 use taxis.
4. Cycle and two-wheeler (60° each), with $\frac{60}{360}\times 216=36$ children each.

(i) Bus (ii) 1/12 (iii) 216 children; none use taxis (iv) cycle and two-wheeler (36 each).

$3\times 4=4\times x$, so $x=3$ days.

Assumptions:

• All workers paint at the same rate.
• The work can be shared equally.
• They work the same number of hours each day.

3 days, assuming every worker works equally fast and the work can be shared.

This is a direct proportion: 3 hours per tank, so $5\times 3=15$ hours.

15 hours

There are $25\times 12=300$ chairs, and $300÷20=15$.

15 rows

The school day is $8\times 45=360$ minutes, and $360÷9=40$.

40 minutes

In one hour they fill $\frac{1}{3}+\frac{1}{2}=\frac{5}{6}$ of the tank.

Time $=1÷\frac{5}{6}=\frac{6}{5}$ hours $=$ 1 hour 12 minutes.

1.2 hours = 1 h 12 min.

$42\times 63=x\times 54$, so $x=\frac{2646}{54}=49$.

49 machines

The distance is $2\times 60=120$ km, and $120÷80=1.5$ h.

1.5 hours (1 h 30 min)