| Image | Width (in mm) | Height (in mm) |
|---|---|---|
| Image A | 60 | 40 |
| Image B | 40 | 20 |
| Image C | 30 | 20 |
| Image D | 90 | 60 |
| Image E | 60 | 60 |
?In-text Question — Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?
Width factor: \(\dfrac{90}{60}=1.5\). Height factor: \(\dfrac{60}{40}=1.5\).
Both dimensions of D changed by the exact same factor (1.5, or \(\tfrac{3}{2}\)) compared to A, so images A and D are proportional — which is why D looks similar to A, just larger.
?In-text Question — By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?
Dividing corresponding terms: \(\dfrac{90}{60}=\dfrac{3}{2}\) and \(\dfrac{60}{40}=\dfrac{3}{2}\) — both terms scale by the same factor, \(\tfrac{3}{2}\) (i.e. 1.5).
?In-text Question — In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below.
______ : ______
Is the teacher-to-student ratio in your school proportional to the one in my school?
______ : ______
Is the teacher-to-student ratio in your school proportional to the one in my school?
The answer depends on your school, so here is the method with a sample.
Ratio in the textbook school \(=5:170=1:34\) [divide both terms by 5].
Sample: suppose your school has 60 teachers and 2040 students.
Ratio \(=60:2040\)
\(=1:34\) [divide both terms by 60]
Both ratios have the same simplest form \(1:34\), so they are proportional.
Now suppose another school has 50 teachers and 1500 students.
Ratio \(=50:1500=1:30\)
\(1:30\ne1:34\), so this ratio is not proportional to \(5:170\).
?In-text Question — Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard?
______ : ______
Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard?
______ : ______
Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard?
The answer depends on your classroom, so here is the method with a sample.
Suppose the blackboard is 240 cm wide and 120 cm high.
Ratio of width to height \(=240:120\)
\(=2:1\) [divide both terms by 120]
A rectangle with width and height in the ratio \(2:1\) is proportional to the blackboard.
Take a width of 8 cm.
Height \(=8\div2=4\) cm
Check: \(8:4=2:1\) ✓
MTMath Talk — Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
Not necessarily the same absolute size — different students may draw their rectangles at different scales. But if every rectangle keeps the same width-to-height ratio as the blackboard, all the rectangles will look similar (same shape), even though their sizes differ, exactly like images A, C, and D from the start of this chapter.
Filter Coffee!
Filter coffee is a beverage made by mixing coffee decoction with milk. Manjunath usually mixes 15 mL of coffee decoction with 35 mL of milk to make one cup of filter coffee in his coffee shop.
In this case, we can say that the ratio of coffee decoction to milk is 15 : 35.
If customers want ‘stronger’ filter coffee, Manjunath mixes 20 mL of decoction with 30 mL of milk. The ratio here is 20 : 30.
MTMath Talk — Why is this coffee stronger?
The regular ratio \(15:35\) in simplest form is \(3:7\), so decoction : milk \(=\tfrac{3}{7}\approx0.43\). For the stronger coffee, \(20:30=2:3\), so decoction : milk \(=\tfrac{2}{3}\approx0.67\). There is more decoction for the same amount of milk than in the regular coffee.
And when they want ‘lighter’ filter coffee, he mixes 10 mL of coffee and 40 mL of milk, making the ratio 10 : 40.
MTMath Talk — Why is this coffee lighter?
For the lighter coffee, \(10:40=1:4\), so decoction : milk \(=\tfrac{1}{4}=0.25\), which is less than the regular \(\tfrac{3}{7}\approx0.43\). There is less decoction for the same amount of milk.
The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee.
| Coffee Decoction (in mL) | Milk (in mL) | Simplest form | Decoction/Milk | Regular/Strong/Light |
|---|---|---|---|---|
| 300 | 600 | 1:2 | 0.5 | Strong |
| 150 | 500 | 3:10 | 0.3 | Light |
| 200 | 400 | 1:2 | 0.5 | Strong |
| 24 | 56 | 3:7 | ≈0.43 | Regular |
| 100 | 300 | 1:3 | ≈0.33 | Light |
Figure it Out — Set 1
Seven questions from pages 165–167 of the textbook.
1Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7 (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12 (vi) 24 : 8 :: 9 : 3
Checking each by cross multiplication (\(ad\) vs. \(bc\)):
(i) \(4\times21=84\), \(7\times12=84\) → equal → True
(ii) \(8\times6=48\), \(3\times24=72\) → unequal → False
(iii) \(7\times7=49\), \(12\times12=144\) → unequal → False
(iv) \(21\times10=210\), \(6\times35=210\) → equal → True
(v) \(12\times12=144\), \(18\times28=504\) → unequal → False
(vi) \(24\times3=72\), \(8\times9=72\) → equal → True
2Give 3 ratios that are proportional to 4 : 9.
Multiply both terms of \(4:9\) by the same number.
Multiply by 2: \(4\times2:9\times2 = 8:18\)
Multiply by 3: \(4\times3:9\times3 = 12:27\)
Multiply by 4: \(4\times4:9\times4 = 16:36\)
3Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______
18:24 simplifies to 3:4. Scaling each given first term to match:
\(3:\_\_\): factor \(=3\div18=\tfrac16\), so second term \(=24\times\tfrac16=4\) → 3:4
\(12:\_\_\): factor \(=12\div18=\tfrac23\), so second term \(=24\times\tfrac23=16\) → 12:16
\(20:\_\_\): factor \(=20\div18=\tfrac{10}{9}\), so second term \(=24\times\tfrac{10}{9}=\tfrac{80}{3}\) → 20: 80/3 (not a whole number)
\(27:\_\_\): factor \(=27\div18=\tfrac32\), so second term \(=24\times\tfrac32=36\) → 27:36
4Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.
Measure the longer side and the shorter side of each rectangle with a scale (the rectangles are drawn at different angles, so compare long side : short side).
A: \(0.6\text{ cm}\) by \(1.8\text{ cm}\) → long : short \(=1.8:0.6=3:1\)
B: \(1.8\text{ cm}\) by \(1.2\text{ cm}\) → \(3:2\)
C: \(5.1\text{ cm}\) by \(2.0\text{ cm}\) → \(\approx 5:2\)
D: \(3.9\text{ cm}\) by \(1.3\text{ cm}\) → \(3:1\)
E: \(1.7\text{ cm}\) by \(0.7\text{ cm}\) → \(\approx 5:2\)
Rectangles with the same ratio are similar: A and D (both \(3:1\)), and C and E (both \(5:2\)). B (\(3:2\)) is not similar to any other.
5Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates’ drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?
Measure the given rectangle: width \(\approx 3.5\text{ cm}\), height \(\approx 2.1\text{ cm}\), so width : height \(\approx 5:3\).
Smaller rectangle: \(2.5\text{ cm}\times 1.5\text{ cm}\) (ratio \(5:3\)). Bigger rectangle: \(5\text{ cm}\times 3\text{ cm}\) (ratio \(5:3\)).
Classmates' rectangles will differ in size from yours, but they are not wrong — as long as everyone's rectangle keeps the same ratio of width to height as the original, it is proportional, regardless of its size.
6The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form.
(a)
(b)
(a) Count the bricks in one repeating block of 5 bricks wide and 3 rows.
Row 1: 2 grey and 3 coloured
Row 2: 3 grey and 2 coloured
Row 3: 4 grey and 1 coloured
Grey bricks \(=2+3+4=9\)
Coloured bricks \(=3+2+1=6\)
Ratio \(=9:6=3:2\)
(b) One repeating block is 4 bricks wide and 7 rows high.
Total bricks in the block \(=4\times7=28\)
Coloured bricks in the 7 rows \(=1+2+2+2+2+2+1=12\)
Grey bricks \(=28-12=16\)
Ratio \(=16:12=4:3\)
7Let us draw some human figures. Measure your friend’s body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— head : torso ______ : ______ torso : arms ______ : ______ torso : legs ______ : ______ Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above.
Try yourself: write head : torso, torso : arms and torso : legs from your measurements in simplest form, then draw a figure whose parts are in the same ratios.
MTMath Talk — Does the drawing look more realistic if the ratios are proportional? Why? Why not?
Yes. Real human bodies have fairly consistent head:torso:limb proportions, so a figure that keeps those same ratios — just scaled to a different overall size — looks like a natural, correctly-proportioned person. A figure where the ratios are changed (e.g. torso stretched but arms not) looks distorted, exactly like the elongated tiger in image B from the start of this chapter.
AActivity — Take your favourite dish. Find out all the ingredients and their respective quantities needed to make the dish for your family. Suppose you are celebrating a festival and you want to invite 15 guests. Find out the quantities of the ingredients required to cook the same dish for them.
The answer depends on your dish. Here is the method, with poha for a family of 4 as a sample.
The number of people changes from 4 to 15.
So every quantity is multiplied by \(\dfrac{15}{4}\).
| Ingredient | For 4 people | Working | For 15 people |
|---|---|---|---|
| Poha | 2 cups | \(2\times\frac{15}{4}\) | 7.5 cups |
| Onion | 1 | \(1\times\frac{15}{4}\) | 3.75, so about 4 onions |
| Oil | 2 tbsp | \(2\times\frac{15}{4}\) | 7.5 tbsp |
| Peanuts | 4 tbsp | \(4\times\frac{15}{4}\) | 15 tbsp |
| Turmeric | \(\frac12\) tsp | \(\frac12\times\frac{15}{4}\) | \(\frac{15}{8}\) tsp, about 2 tsp |
Figure it Out — Set 2
Two questions from pages 170–171 of the textbook.
1The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
Take 52 weeks in a year.
In 52 weeks the Earth travels 940,000,000 km.
In 1 week it travels \(x\) km.
\(940{,}000{,}000:52::x:1\)
\(x=\dfrac{940{,}000{,}000}{52}\)
\(x\approx18{,}076{,}923\)
2A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of 10-feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness.
Bricks required for a 10 ft wall \(=1450\), so the ratio is \(10:1450\).
Find the length of every wall.
Left wall \(=12\) ft
Inner wall \(=12\) ft
Right wall \(=12\) ft
Top wall of the left room \(=9\) ft
Top wall of the right room \(=15\) ft
Bottom wall of the left room \(=9\) ft
Bottom wall of the right room \(=15\) ft (this includes the 6 ft over the small room)
Walls of the small room: left \(=9\) ft, right \(=9\) ft, bottom \(=6\) ft
Total length of walls \(=12+12+12+9+15+9+15+9+9+6=108\) ft
Using \(10:1450::108:x\): \(x=\dfrac{1450\times108}{10}=15{,}660\).
Note: the textbook’s own list of walls shows 15 ft twice, and counts the 6 ft top of the small room as a wall. We follow the textbook. If that 6 ft were left open, the total would be 102 ft and the answer 14,790 bricks.
MTMath Talk — Puneeth’s father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion —
50 : 2 :: 75 : __
Would it take Puneeth’s father more time or less time to reach Kanpur? Think about it.
50 : 2 :: 75 : __
Would it take Puneeth’s father more time or less time to reach Kanpur? Think about it.
Distance from Lucknow to Kanpur \(=50\times2=100\) km.
Time at 75 km/h \(=\dfrac{100}{75}\)
\(=\dfrac{4}{3}\) hours
\(=1\) hour 20 minutes
This is less than 2 hours, so he takes less time.
Now try the proportion \(50:2::75:x\).
\(x=\dfrac{2\times75}{50}=3\) hours
3 hours is more than 2 hours, which cannot be right when the speed increases.
Here the speed and the time change in opposite ways: when one goes up, the other goes down.
AActivity — Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo and create a table like the one given below. See if the volume of shampoo is proportional to the price.Container Volume Price Sachet 6 mL ₹2 Small Bottle 180 mL ₹154 Medium Bottle 340 mL ₹276 Large Bottle 1000 mL ₹540
| Container | Volume | Price |
|---|---|---|
| Sachet | 6 mL | ₹2 |
| Small Bottle | 180 mL | ₹154 |
| Medium Bottle | 340 mL | ₹276 |
| Large Bottle | 1000 mL | ₹540 |
If the volume is proportional to the price, the price per mL is the same for every container.
| Container | Volume | Price | Price per mL |
|---|---|---|---|
| Sachet | 6 mL | ₹2 | \(2\div6\approx\)₹0.33 |
| Small Bottle | 180 mL | ₹154 | \(154\div180\approx\)₹0.86 |
| Medium Bottle | 340 mL | ₹276 | \(276\div340\approx\)₹0.81 |
| Large Bottle | 1000 mL | ₹540 | \(540\div1000=\)₹0.54 |
The price per mL is different for different containers.
Compare the sachet and the small bottle: volumes \(6:180=1:30\), prices \(2:154=1:77\).
\(1:30\ne1:77\), so these ratios are not proportional.
MTMath Talk — Why do you think that the ratio of the prices is not proportional to the ratio of the volumes? Discuss the pros and cons of different size bottles for the company and for customers. For reducing ecological footprint, what would you recommend to the company and to the customer?
Does the same occur for other products?
Make similar tables for other products in the market, capturing different prices for different measures of the same product, e.g., rice or atta (flour).
Observe the products for which the prices are proportional to the different measures.
Discuss in class the proportionality of prices to measures of the same product.
Does the same occur for other products?
Make similar tables for other products in the market, capturing different prices for different measures of the same product, e.g., rice or atta (flour).
Observe the products for which the prices are proportional to the different measures.
Discuss in class the proportionality of prices to measures of the same product.
The ratio of the volumes of a sachet to a small bottle is \(6:180=1:30\). The ratio of their prices is \(2:154=1:77\). Since \(1:30\ne1:77\), the ratios are not proportional — the price per mL is not the same for every container size (about ₹0.33, ₹0.86, ₹0.81 and ₹0.54 per mL for the sachet, small, medium and large containers).
Larger containers usually reduce packaging waste per mL of product (better for the environment) and, among the bottles, give customers a lower per-mL cost, but require more money upfront, which can be a barrier for some customers; smaller sachets let customers try a product or manage tight budgets, but they create more packaging waste overall. A reasonable recommendation is for companies to price larger sizes proportionally cheaper per unit (to reward bulk buying and reduce waste), while still offering low-cost trial sizes for accessibility.
Other products (sample prices; real prices change from shop to shop):
| Product | Quantity | Price | Price per kg |
|---|---|---|---|
| Packed rice | 1 kg | ₹60 | ₹60 |
| Packed rice | 5 kg | ₹290 | ₹58 |
| Packed rice | 10 kg | ₹560 | ₹56 |
| Loose tomatoes | 500 g | ₹20 | ₹40 |
| Loose tomatoes | 1 kg | ₹40 | ₹40 |
| Loose tomatoes | 2 kg | ₹80 | ₹40 |
Packed rice: the price per kg falls as the pack gets bigger, so price is not proportional to quantity.
Loose tomatoes: the price per kg is always ₹40, so price is proportional to quantity.
AActivity — Form a pair. Collect 12 countable objects or counters (it can be coins, seeds, or pebbles). Now, share them between the two of you in different ways.
If you divide them equally, what is the ratio of the number of counters with each of you?
If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
Now, if you want to share the counters between the two of you in the ratio of 3 : 1, how many counters would each of you get? Share the counters in different ways and see which combination is in the ratio 3 : 1.
If you divide them equally, what is the ratio of the number of counters with each of you?
If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
Now, if you want to share the counters between the two of you in the ratio of 3 : 1, how many counters would each of you get? Share the counters in different ways and see which combination is in the ratio 3 : 1.
Equal sharing
Each of you gets \(12\div2=6\) counters.
Ratio \(=6:6=1:1\)
Partner gets 5
You get \(12-5=7\) counters.
Ratio of your partner’s counters to yours \(=5:7\)
Sharing in the ratio 3 : 1
Number of equal parts \(=3+1=4\)
Counters in each part \(=12\div4=3\)
Partner gets \(3\times3=9\) counters.
You get \(1\times3=3\) counters.
Check: \(9+3=12\) and \(9:3=3:1\) ✓
Figure it Out — Set 3
Five questions from page 175 of the textbook.
1Divide ₹4,500 into two parts in the ratio 2 : 3.
Total number of parts \(=2+3=5\).
Value of each part \(=4500\div5=900\).
First part \(=2\times900=1800\); second part \(=3\times900=2700\).
2In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 mL of the solution, how much acid and water does the solution contain?
Total number of parts \(=1+5=6\).
Value of each part \(=240\div6=40\) mL.
Acid \(=1\times40=40\) mL; water \(=5\times40=200\) mL.
3Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 mL of yellow to the mixture. What is the new ratio of blue and yellow in the paint?
Total number of parts \(=3+5=8\).
Value of each part \(=40\div8=5\) mL.
Blue \(=3\times5=15\) mL; yellow \(=5\times5=25\) mL.
Adding 20 mL more yellow: new yellow \(=25+20=45\) mL. New ratio blue:yellow \(=15:45=1:3\).
4To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?
Total number of parts \(=2+1=3\).
Value of each part \(=6\div3=2\) cups.
Rice \(=2\times2=4\) cups; urad dal \(=1\times2=2\) cups.
5I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?
Total number of parts \(=3+5=8\), so each part \(=\dfrac18\) of a bucket.
Red \(=3\times\dfrac18=\dfrac38\) of a bucket; yellow \(=5\times\dfrac18=\dfrac58\) of a bucket.
Adding another bucket of yellow: new yellow \(=\dfrac58+1=\dfrac{13}{8}\) of a bucket.
New ratio red:yellow \(=\dfrac38:\dfrac{13}{8}=3:13\).
Figure it Out — Set 4
Twelve questions from pages 176–177 of the textbook.
1Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.
HCF of 600 and 900 is 300. \(600:900 = 2:3\).
2Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?
Assume the 204 people this year also include the teachers, as the 162 did last year.
Capacity per bus \(=162\div3=54\). Using \(3:162::x:204\): \(x=\dfrac{3\times204}{162}=\dfrac{34}{9}\approx3.78\), so 4 buses are needed (round up, since a fraction of a bus isn't possible).
Total capacity in 4 buses \(=4\times54=216\); with only 204 students, \(216-204=12\) seats will be empty.
3The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?
Population density (people per sq. km): Delhi \(=\dfrac{30{,}000{,}000}{1484}\approx20{,}216\). Mumbai \(=\dfrac{20{,}000{,}000}{550}\approx36{,}364\).
4A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?
Crane: height \(=155\) cm, neck : rest of body \(=4:6\).
Total number of parts \(=4+6=10\).
Value of each part \(=155\div10=15.5\) cm.
Neck of the crane \(=4\times15.5=62\) cm.
Rest of the body of the crane \(=6\times15.5=93\) cm.
Check: \(62+93=155\) cm ✓
Now take my height as 165 cm, with the same ratio \(4:6\).
Total number of parts \(=4+6=10\).
Value of each part \(=165\div10=16.5\) cm.
Neck \(=4\times16.5=66\) cm.
Rest of the body \(=6\times16.5=99\) cm.
Check: \(66+99=165\) cm ✓
Shortcut: neck \(=\dfrac{4}{10}\times\text{height}=\dfrac{4}{10}\times165=66\) cm.
For a height of 165 cm: neck = 66 cm, rest of body = 99 cm.
5Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2½ palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”
Setting up \(2.5:\tfrac37::x:9\), cross multiplying: \(2.5\times9=x\times\tfrac37\), so \(x=\dfrac{2.5\times9\times7}{3}=\dfrac{157.5}{3}=52.5\).
6Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2?
Given: Harmain’s age is 1 and her brother’s age is 5. Let the ratio of Harmain’s age to her brother’s age be \(1:2\) after \(x\) years.
\(x+1:x+5=1:2\), so \(\dfrac{x+1}{x+5}=\dfrac12\). Cross multiplying: \(2x+2=x+5\), so \(x=3\).
After 3 years, Harmain will be \(1+3=4\) and her brother \(5+3=8\) — ratio \(4:8=1:2\). ✓
7The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?
\(37:2::x:1\), so \(x=\dfrac{37\times1}{2}=18.5\).
8It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).
1 tonne \(=1000\) kg, so 10 tonnes \(=10{,}000\) kg. Also, 1 acre \(=43{,}560\) sq. ft.
The ratio of cow manure to area of land is \(10{,}000:43{,}560\). Size of the plot \(=200\times500=100{,}000\) sq. ft.
Let the cow manure required be \(x\) kg. Then \(x:100{,}000::10{,}000:43{,}560\), so \(x=\dfrac{100{,}000\times10{,}000}{43{,}560}\approx22{,}956.8\) kg.
9A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?
10 litres \(=10{,}000\) mL. \(500:15::10{,}000:x\), so \(x=\dfrac{15\times10{,}000}{500}=300\) seconds \(=5\) minutes.
10One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?
1 acre \(=43{,}560\) sq. ft, costing ₹15,00,000. \(43{,}560:15{,}00{,}000::2{,}400:x\), so \(x=\dfrac{15{,}00{,}000\times2{,}400}{43{,}560}\approx82{,}645\).
11A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?
Ratio of the time taken to plough by tractor and oxen \(=1:4\).
Time taken by a pair of oxen to plough 1 acre \(=6\) hours. Time taken by a tractor to plough 1 acre \(=\dfrac64=1.5\) hours.
Time taken by a pair of oxen to plough 20 acres \(=20\times6=120\) hours. Time taken by the tractor to plough 20 acres \(=20\times1.5=30\) hours.
12Try This — The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?
Total number of parts \(=3+1=4\).
Mass of each part \(=7.74\div4=1.935\) g.
Copper \(=3\times1.935=5.805\) g; nickel \(=1\times1.935=1.935\) g.
Cost of 1000 g copper \(=\)₹906, so cost of 5.805 g copper \(=\dfrac{906}{1000}\times5.805\approx\)₹5.26.
Cost of 1000 g nickel \(=\)₹1,341, so cost of 1.935 g nickel \(=\dfrac{1341}{1000}\times1.935\approx\)₹2.59.
Puzzle Time — Binairo
Binairo
Binairo, also known as Takuzu, is a logic puzzle with simple rules. Binairo is generally played on a square grid with no particular size. Some cells start out filled with two symbols: here horizontal and vertical lines. The rest of the cells are empty. The task is to fill cells in such a way that:
- Each row and each column must contain an equal number of horizontal and vertical lines.
- More than two horizontal or vertical lines can’t be adjacent.
- Each row is unique. Each column is unique.
Solve the following Binairo puzzles:
Dot Grid
Dot Grid
The textbook gives two blank dot-grid sheets at the end of the chapter, to be used for the drawing activities (for example, enlarging and reducing figures). Print this page and use the grid below.
Continue with Part 2, Chapter 1
Move on to Fractions in Disguise next.