64 worked answers · short steps first, explanations when you need them
7.1–7.4 Ratios and proportional change
Textbook pages 160–165
Question similar-image-factors
Images A, C and D have dimensions 60×40 mm, 30×20 mm and 90×60 mm. Why are they similar, while B (40×20) and E (60×60) are distorted?
Steps only
A:C width and height factors are both 1/2.
A:D factors are both 3/2.
B width factor is 2/3 but height factor is 1/2; E also changes the dimensions by unequal factors.
Answer
A, C and D are similar because width and height are scaled by the same factor. B and E are distorted because their scale factors differ.
Question simplify-image-ratios
Reduce the ratios for images A–E with dimensions A 60:40, B 40:20, C 30:20, D 90:60 and E 60:60.
Steps only
Divide each pair by its greatest common factor.
Answer
A=3:2, B=2:1, C=3:2, D=3:2, E=1:1. A, C and D are proportional; B and E are not.
Question true-proportions
Which 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?
Steps only
Reduce both ratios in each statement or compare cross-products.
(i) 4/7=12/21; (iv) 21/6=35/10; (vi) 24/8=9/3.
Answer
Statements (i), (iv) and (vi) are true.
Question ratios-to-4-9
Give three ratios proportional to 4:9.
Steps only
Multiply both terms by the same number.
Using 2, 3 and 4 gives 8:18, 12:27 and 16:36.
Answer
Examples: 8:18, 12:27 and 16:36.
Question missing-18-24
Complete 3:__, 12:__, 20:__ and 27:__.
Steps only
Simplify 18:24 to 3:4.
Use the same scale factor on both terms.
Answer
3:4; 12:16; 20:80/3; 27:36.
Question similar-rectangles
Which of the five rectangles A–E in the figure are similar? You can check by measuring their widths and heights and comparing the ratios.
Steps only
Measure width:height for each rectangle.
Rectangles C, D and E have the same simplified ratio.
Answer
C, D and E are similar; A and B have different proportions.
Question same-ratio-drawings
Can two students draw different-sized rectangles that have the same width-to-height ratio? Are either necessarily wrong?
Steps only
Start with a width and height in a fixed ratio.
Scale both dimensions by the same factor to make a smaller or larger rectangle.
Answer
Yes. They may have different sizes but the same shape and ratio, so both can be correct.
Question brick-grey-colour-a
In pattern (a), each repeated block has 9 grey and 6 coloured bricks. Find the simplest ratio.
Steps only
Write grey:coloured=9:6.
Divide both terms by 3.
Answer
3:2.
Question brick-grey-colour-b
In pattern (b), each block has 16 grey and 12 coloured bricks. Find the simplest ratio.
Steps only
Write 16:12.
Divide both terms by 4.
Answer
4:3.
Question human-proportions
How should a student use measured head, torso, arm and leg lengths to draw a proportional figure?
Steps only
Measure the chosen parts and record ratios such as head:torso and torso:legs.
Choose a convenient scale factor and multiply every measured length by it.
Answer
Keep all corresponding body-part lengths in the same scale; actual measurements vary from person to person.
Question mother-child-age
When Neelima is 3, her mother is 30. Compare their age ratio now with the ratio when Neelima is 12.
Steps only
At age 3: 3:30=1:10.
Nine years later, the mother is 39; 12:39=4:13.
Answer
The ratios are 1:10 and 4:13, so they are not proportional.
Question missing-14-21-ratios
Complete the ratios proportional to 14:21: (i) __:42, (ii) 6:__, and (iii) 2:__.
Steps only
(i) 42÷21=2, so 14×2=28.
(ii) 6÷14=3/7, so 21×3/7=9.
(iii) 14÷7=2, so 21÷7=3.
Answer
(i) 28:42;
(ii) 6:9;
(iii) 2:3.
Question coffee-strength-ratios
Regular coffee uses 15 mL decoction and 35 mL milk. Stronger coffee uses 20 mL decoction and 30 mL milk; lighter coffee uses 10 mL and 40 mL. Simplify each decoction:milk ratio and compare the strengths.
Steps only
Regular: 15:35=3:7; decoction is 15/50=30% of the cup.
Stronger: 20:30=2:3; decoction is 20/50=40%.
Lighter: 10:40=1:4; decoction is 10/50=20%.
Answer
Regular=3:7, stronger=2:3 and lighter=1:4. The 2:3 mixture is strongest and the 1:4 mixture is lightest.
Question proportionality-examples-3-4
Nitin builds a 60 ft wall using 3 bags of cement. Hari builds a 40 ft wall using 2 bags. Are their cement amounts proportional to wall length? Then compare the example school ratio of 5 teachers to 170 students with your own school.
Steps only
Nitin: 60:3=20:1. Hari: 40:2=20:1.
For your school, write teachers:students as T:S, then simplify both T:S and 5:170=1:34.
Answer
The cement use is proportional: both walls use 1 bag per 20 ft. The school comparison depends on your school’s counts; compare its simplified teachers:students ratio with 1:34.
Question filter-coffee-table
The regular coffee ratio is 15:35 (decoction:milk). For each mixture in the book’s table, simplify its decoction:milk ratio and compare it with the regular ratio. Which has more decoction per amount of milk?
Divide the new first quantity by the old first quantity.
Multiply the second quantity by the same factor if the quantities are directly proportional.
Answer
Factor of change = new first quantity ÷ old first quantity; use that same factor on the other term in a proportion.
Question fractional-ratio-factor
What factor should we multiply 14 by to get 6? Must the factor be a whole number?
Steps only
6÷14=3/7.
A scale factor can be a fraction.
Answer
Multiply by 3/7; the factor need not be a whole number.
Question image-d-scale-factor
Image A is 60 mm wide and 40 mm high. Image D is 90 mm wide and 60 mm high. By what factor do D’s width and height change from A?
Steps only
Width factor=90÷60=3/2.
Height factor=60÷40=3/2.
Answer
Both dimensions are multiplied by 3/2.
Question scale-image-a-to-d
By what factor must the ratio 60:40 for image A be multiplied to get image D’s ratio 90:60?
Steps only
90÷60=3/2 and 60÷40=3/2.
Multiply both terms by 3/2.
Answer
Multiply both terms by 3/2: 60:40 becomes 90:60.
Question image-b-e-simplest-ratios
Image B measures 40×20 mm and image E measures 60×60 mm. Find each width:height ratio in simplest form.
Steps only
B:40:20=2:1.
E:60:60=1:1.
Answer
Image B has ratio 2:1; image E has ratio 1:1.
7.5 Solving direct-proportion problems
Textbook pages 166–175
Question lemonade-sugar
A recipe uses 10 spoons of sugar for 6 glasses of lemonade. If 18 more glasses are needed at the same sweetness, how much sugar should be added?
Steps only
Total glasses=6+18=24.
24÷6=4 times as many glasses.
10×4=40 spoons.
Answer
Add 40 spoons of sugar for 18 more glasses (24 glasses in total).
Question cement-wall-strength
A 60 ft wall uses 3 cement bags; a 40 ft wall uses 2. Is the cement use proportional to wall length?
Steps only
60:3=20:1.
40:2=20:1.
Answer
Yes. Both use 1 bag per 20 ft, so the cement-to-length ratios match.
Question teachers-students
The example school has 5 teachers and 170 students. How should you compare your school with it?
Steps only
Write your school’s measured counts as teachers:students.
Simplify both your ratio and 5:170=1:34.
Answer
They are proportional only if your simplified ratio is also 1:34.
Question blackboard-ratio
How can you draw a smaller rectangle with the same width-to-height ratio as a classroom blackboard?
Steps only
Measure its width w and height h.
Choose a scale factor k and draw dimensions kw and kh.
Answer
Use any common scale factor on both dimensions; for example, halve both measurements.
Question earth-week-distance
Earth travels about 940 million km around the Sun in 52 weeks. How far does it travel in one week on average?
Steps only
Divide the yearly distance by 52.
940,000,000÷52≈18,076,923.08.
Answer
About 18.08 million km per week (approximately 18,076,923 km).
Question mason-bricks-house
A 10 ft wall needs about 1450 bricks. The shown house has wall lengths 12,12,12,15,9,15,9,9,9 and 6 ft. Estimate the total bricks.
Steps only
Add wall lengths: total=108 ft.
At 1450 bricks per 10 ft: 108/10×1450=15,660.
Answer
About 15,660 bricks.
Question speed-time-inverse
A journey takes 2 h at 50 km/h. How long at 75 km/h?
Steps only
The distance is 50×2=100 km.
Time=distance÷speed=100÷75=4/3 h.
Answer
1 hour 20 minutes. This is inverse proportion: for a fixed journey, higher speed means less time.
Question midday-meal-rice
A cook uses 15 kg of rice for 120 students. How much rice is needed for 80 students if the same amount is served per student?
Steps only
80/120=2/3.
15×2/3=10 kg.
Answer
The cook should use 10 kg of rice.
Question car-distance-four-hours
A car travels 90 km in 150 minutes at a steady speed. How far will it travel in 4 hours?
Steps only
4 hours=240 minutes.
Scale factor=240/150=8/5.
Distance=90×8/5=144 km.
Answer
The car travels 144 km in 4 hours.
Question shampoo-sample-table
A 6 mL shampoo sachet costs ₹2 and a 180 mL bottle costs ₹154. Check if price is proportional to volume. What should a shopper consider when comparing container sizes?
Steps only
Volume ratio 6:180=1:30; price ratio 2:154=1:77.
Since 1:30≠1:77, the quantities are not proportional.
Sachet price per mL=₹2/6; bottle price per mL=₹154/180.
Answer
No. Price is not proportional to volume. Here the sachet is cheaper per mL (about ₹0.333) than the bottle (about ₹0.856).
Question student-count-factor
A school’s student count changes from 120 to 80. What is the factor of change in the first quantity?
Steps only
80÷120=2/3.
Answer
The factor of change is 2/3.
7.6–7.7 Ratios in sharing and mixtures
Textbook pages 175–176
Question split-4500
Divide ₹4,500 in the ratio 2:3.
Steps only
There are 2+3=5 equal ratio-parts.
One part is ₹4,500÷5=₹900.
The shares are 2×900 and 3×900.
Answer
₹1,800 and ₹2,700.
Question acid-water
A 240 mL solution has acid:water=1:5. Find each volume.
Steps only
Total parts=1+5=6.
One part=240÷6=40 mL.
Answer
Acid=40 mL; water=200 mL.
Question blue-yellow-green
Blue:yellow=3:5 in 40 mL green paint. Find each amount, then add 20 mL yellow and give the new ratio.
Steps only
Total parts=8; each part=40÷8=5 mL.
Blue=15 mL, yellow=25 mL.
After adding 20 mL yellow, yellow=45 mL; 15:45 simplifies to 1:3.
Answer
Initially blue=15 mL and yellow=25 mL. New blue:yellow=1:3.
Question idli-rice-dal
Rice:urad dal=2:1. How much of each is needed for 6 cups total?
Steps only
Total parts=3.
Each part=6÷3=2 cups.
Answer
Rice=4 cups and urad dal=2 cups.
Question bucket-yellow-paint
A bucket contains red:yellow=3:5. One more bucket of yellow paint of the same size is added. Find the new ratio.
Steps only
Let the original bucket contain 3k red and 5k yellow, so its total is 8k.
The added bucket contributes 8k more yellow; total yellow=13k.
Answer
Red:yellow=3:13.
Question orange-apple-juice
Anagh mixes 600 mL orange juice and 900 mL apple juice. Simplify orange:apple.
Steps only
600:900.
Divide by 300.
Answer
2:3.
Question school-buses
Three full buses carry 162 people. How many buses are needed for 204 people, and how many seats are empty?
Steps only
Capacity per bus=162÷3=54.
204÷54 requires 4 buses.
Four buses hold 216, so 216−204=12 seats remain.
Answer
4 buses; 12 seats remain empty.
Question city-density
Delhi has 30 million people over 1,484 km²; Mumbai has 20 million over 550 km². Which is more crowded?
Steps only
Delhi: 30,000,000÷1,484≈20,216 people/km².
Mumbai: 20,000,000÷550≈36,364 people/km².
Answer
Mumbai is more crowded by population density.
Question crane-neck
A 155 cm crane has neck:rest=4:6. If a person of height H has the same ratio, how long is the neck?
Steps only
Total ratio parts=10.
Neck fraction=4/10=2/5.
Answer
Neck length=2H/5=0.4H.
Question saffron-rate
2½ palas of saffron cost 3/7 niskas. How many palas can be bought for 9 niskas?
Steps only
9 niskas is 9÷(3/7)=21 times the cost.
Multiply the saffron amount by 21: 2½×21.
Answer
52½ palas.
Question harmain-age-ratio
Harmain is 1 and her brother is 5. After how many years will her age:his age be 1:2?
Steps only
After x years, ages are 1+x and 5+x.
(x+1)/(x+5)=1/2, so 2x+2=x+5.
x=3.
Answer
After 3 years Harmain will be 4 and her brother 8; the ratio is 1:2.
Question gold-mass
Equal volumes of gold and water have masses in ratio 37:2. If 1 L water has mass 1 kg, find the mass of 1 L gold.
Steps only
37:2=x:1.
x=37/2=18.5.
Answer
18.5 kg.
Question manure-area
Use 10 tonnes of manure per acre. A field is 200 ft by 500 ft. How much manure is needed?
Steps only
Field area=200×500=100,000 ft².
1 acre=43,560 ft².
Manure=10×100,000/43,560≈22.9568 tonnes.
Answer
About 22.96 tonnes (approximately 22,957 kg).
Question tap-bucket
A tap fills 500 mL in 15 seconds. How long for 10 L at the same flow rate?
Steps only
10 L=10,000 mL, which is 20 times 500 mL.
Time=15×20=300 seconds.
Answer
300 seconds, or 5 minutes.
Question land-cost
One acre costs ₹15,00,000. Estimate the cost of 2,400 ft² if 1 acre=43,560 ft².
Steps only
Cost=₹15,00,000×2,400/43,560.
Answer
Approximately ₹82,645.
Question tractor-plough-time
Oxen take 6 hours per acre. A tractor works four times faster. How long for 20 acres with each?
Steps only
Oxen: 20×6=120 h.
Tractor time per acre=6÷4=1.5 h; for 20 acres, 30 h.
Answer
Oxen: 120 hours; tractor: 30 hours.
Question coin-alloy-cost
A 7.74 g coin has copper:nickel=3:1. Copper costs ₹906/kg and nickel ₹1,341/kg. Find their combined metal cost.
Steps only
Total parts=4; copper=7.74×3/4=5.805 g; nickel=1.935 g.
Copper cost=₹0.906×5.805≈₹5.26.
Nickel cost=₹1.341×1.935≈₹2.59.
Answer
Combined metal cost≈₹7.85 (about ₹5.26 copper and ₹2.59 nickel).
Question share-42-counters
Divide 42 counters in the ratio 4:3. Then write the general rule for splitting x in the ratio m:n.
Steps only
4+3=7 total parts; 42÷7=6 per part.
First share: 4×6=24; second share: 3×6=18.
For x in m:n: first=m x/(m+n), second=n x/(m+n).
Answer
The shares are 24 and 18. In general they are mx/(m+n) and nx/(m+n).
Question share-business-profit
Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. They share a ₹4,000 profit in the same ratio as their investments. Find each share.
Prashanti receives ₹3,000 and Bhuvan receives ₹1,000.
Question sand-cement-mixture-adjustment
A 40 kg mixture has sand:cement=3:1. How much cement should be added to make the ratio 5:2?
Steps only
Original parts: 3+1=4; one part=40÷4=10 kg.
Sand=30 kg; cement=10 kg.
At 5:2, 30 kg sand requires 30×2/5=12 kg cement.
Add 12−10=2 kg cement.
Answer
Add 2 kg of cement.
Question tea-price-proportionality
One tea costs ₹200 for 200 g. Another costs ₹800 for 1 kg. Are their weight:price ratios proportional? Which tea costs more per kilogram?
Steps only
First: 200:200=1:1. Second: 1000:800=5:4.
First tea: ₹200×5=₹1,000 per kg; second: ₹800 per kg.
Answer
The ratios are not proportional. The ₹200-for-200-g tea is more expensive at ₹1,000/kg versus ₹800/kg.
Question shampoo-volume-price-proportion
A 6 mL sachet costs ₹2 and a 180 mL bottle costs ₹154. Are volume and price proportional?
Steps only
Volume ratio: 6:180=1:30.
Price ratio: 2:154=1:77.
Cross-products: 6×154=924; 180×2=360.
Answer
No. 1:30 and 1:77 are different ratios, so volume and price are not proportional.
Question scale-recipe-for-guests
A recipe gives ingredient amounts for a family of f people. How should the amounts change when cooking the same dish for f+15 people?
Steps only
Scale factor=(f+15)/f.
New amount of each ingredient=(original amount)×(f+15)/f.
Answer
Multiply every original ingredient quantity by (f+15)/f; substitute your family size and recipe amounts.
Question equal-ratio-general-rule
How can a total quantity x be divided into two parts in the ratio m:n?
Steps only
Total equal groups=m+n.
One group=x/(m+n).
Parts=m×x/(m+n) and n×x/(m+n).
Answer
The shares are mx/(m+n) and nx/(m+n).
Question shampoo-environment-choice
Shampoo prices do not rise in the same proportion as container volumes. What should customers and companies consider when choosing package sizes?
Steps only
Compare price per millilitre, not just the total price.
Consider packaging waste, transport, storage and whether the product will be used before it expires.
Answer
Choose using both unit price and waste: a larger bottle may cost less per millilitre, while a small pack can avoid unused product but uses more packaging per millilitre. Companies can reduce material and make packaging recyclable or refillable.
Question share-12-equally
Share 12 counters equally between two people. What is the ratio of their shares?
Steps only
12÷2=6 counters each.
6:6=1:1.
Answer
Each person gets 6 counters, so the ratio is 1:1.
Question share-12-five
There are 12 counters. If your partner gets 5, how many do you get? What is the ratio of your share to your partner’s?
Steps only
12−5=7 counters.
Your share : partner’s share=7:5.
Answer
You get 7 counters; the ratio is 7:5.
Question share-12-three-one
Share 12 counters between two people in the ratio 3:1.
Steps only
3+1=4 equal parts.
12÷4=3 counters per part.
Shares: 9 and 3.
Answer
The shares are 9 counters and 3 counters.
Puzzle Time: Binairo
Textbook pages 178
Question binairo-1
Complete the 6×6 Binairo grid. Given clues: Place vertical lines at row 2 column 3, row 4 columns 3 and 5, and row 6 column 3. Place horizontal lines at row 2 columns 5 and 6, row 3 column 2, and row 5 columns 1 and 5.
Steps only
Use V for a vertical line and H for a horizontal line.
Each row and column needs three of each; no three alike can touch; rows and columns must be distinct.
Complete the 6×6 Binairo grid. Given clues: Place horizontal lines at row 1 column 1, row 2 columns 1 and 6, and row 5 column 6. Place vertical lines at row 3 column 4, row 4 columns 2 and 3, and row 6 columns 2 and 5.
Steps only
Use V for a vertical line and H for a horizontal line.
Each row and column needs three of each; no three alike can touch; rows and columns must be distinct.
Complete the 6×6 Binairo grid. Given clues: Place horizontal lines at row 1 column 5, row 2 column 6, and row 3 columns 1, 3 and 6. Place vertical lines at row 5 columns 3 and 4, and row 6 column 1.
Steps only
Use V for a vertical line and H for a horizontal line.
Each row and column needs three of each; no three alike can touch; rows and columns must be distinct.