Proportional Reasoning-1 Class 8 Solutions

Images A, C, and D are similar because their width and height change by the same factor. B is stretched and E is compressed because their width and height do not scale by one common factor.

Image A is (60\text{ mm}\times40\text{ mm}), while D is (90\text{ mm}\times60\text{ mm}). Both dimensions are multiplied by (\\frac{90}{60}=\\frac{60}{40}=\\frac32), so D is proportional to A.

For B, (40:20=2:1). For E, (60:60=1:1). Neither equals the (3:2) ratio of A, C, and D.

Yes. The HCF of 72 and 96 is 24, and (72:96=3:4). Equivalently, (3\times96=4\times72=288).

The number of glasses is multiplied by (18/6=3), so multiply the sugar by the same factor: (10\times3=30) spoons.

No. (60:3=20:1) and (40:2=20:1). Both use one bag per 20 ft, so the cement use is proportional and the walls have the same stated strength.

Let your school have (T) teachers and (S) students. Reduce (T:S), or test (170T=5S). The ratios are proportional exactly when this equality holds. The numerical result depends on your school data.

Measure blackboard width (W) and height (H). Choose any convenient scale factor (k), then draw width (kW) and height (kH). Different values of (k) give different-sized but similar rectangles.

Initially the ratio is (3:30=1:10). Nine years later it is (12:39=4:13). Adding the same number to both terms does not generally preserve a ratio.

Since (14:21=2:3), the completed ratios are (28:42), (6:9), and (2:3).

Regular coffee has decoction-to-milk ratio (15:35=3:7). The rows are: (300:600=1:2), stronger; (150:500=3:10), lighter; (200:400=1:2), stronger; (24:56=3:7), regular; (100:300=1:3), lighter.

Statements (i), (iv), and (vi) are true. Their cross-products match: (4\times21=7\times12), (21\times10=6\times35), and (24\times3=8\times9).

Multiplying both terms by 2, 3, and 4 gives (8:18, 12:27,) and (16:36).

Since (18:24=3:4), the missing terms are (4, 16, \frac{80}{3},) and (36), respectively.

A, C, and D are similar, allowing for rotation: their longer-side to shorter-side ratios are equal. B and E have different aspect ratios. Measuring the printed sides verifies the comparison.

If the printed rectangle measures (W:H), examples are (\\frac12W:\frac12H) and (2W:2H). Classmates may use other common scale factors, so their rectangles can differ in size without being wrong.

In (a), one repeating block has 9 grey and 6 coloured bricks, so the ratio is (9:6=3:2). In (b), a repeating block has 16 grey and 12 coloured bricks, so the ratio is (16:12=4:3).

Record head:torso, torso:arms, and torso:legs. Choose one scale factor (k) and multiply every measured length by (k). The drawing preserves all three ratios only when the same (k) is used throughout.

The scale factor is (80/120=2/3). Therefore the rice needed is (15\times\frac23=10) kg.

Proportional ratios give (\\frac ca=\frac db). Multiplying by (ab) yields (bc=ad). Therefore, when (a\ne0), the fourth term is (d=\frac{bc}{a}).

Convert 4 hours to 240 minutes. Then (150:90::240:x), so (x=\frac{240\times90}{150}=144) km.

Use the same mass unit. The ratios are (200:200=1:1) and (1000:800=5:4), so they are not proportional. One kilogram of the first tea costs (5\times₹200=₹1000), versus ₹800 for the second; the first is more expensive.

If an ingredient quantity (q) serves (f) people, the amount for 15 guests is (q\times\frac{15}{f}). Apply the same factor to every ingredient so the recipe proportions remain unchanged.

Taking 52 weeks in a year, (\\frac{940{,}000{,}000}{52}\approx18{,}076{,}923) km, or about 18.08 million km.

The outer and inner wall lengths total (108) ft. At (1450/10=145) bricks per foot, the requirement is (108\times145=15{,}660) bricks.

The relation is inverse, so (50:2::75:t) is not valid. The distance is (50\times2=100) km, and the new time is (100/75=4/3) hour, or 1 hour 20 minutes.

No. The mL per rupee values are (3, 180/154\approx1.17, 340/276\approx1.23,) and (1000/540\approx1.85). The ratios differ, and the sachet gives the most shampoo per rupee in this sample, although it creates much more packaging waste.

Equal sharing gives (6:6=1:1). If one person receives 5, the other receives 7, so the ratio is (5:7).

For 12, there are (3+1=4) units, each worth 3, so the shares are 9 and 3. For 42, there are (4+3=7) units, each worth 6, so the shares are 24 and 18.

The investment ratio is (75{,}000:25{,}000=3:1). Four equal units make ₹4,000, so each unit is ₹1,000. Prashanti receives ₹3,000 and Bhuvan ₹1,000.

Initially there are 30 kg sand and 10 kg cement. Keeping sand at 30 kg, (5:2::30:c) gives (c=12) kg. Add (12-10=2) kg cement.

There are 5 ratio units, each worth (₹4500/5=₹900). The parts are (2\times₹900=₹1800) and (3\times₹900=₹2700).

There are 6 units, each (240/6=40) mL. Acid is 40 mL and water is (5\times40=200) mL.

Eight units make 40 mL, so blue is 15 mL and yellow is 25 mL. After adding 20 mL yellow, the ratio is (15:45=1:3).

Three units make 6 cups, so each unit is 2 cups. Use 4 cups rice and 2 cups urad dal.

The original red and yellow amounts are (3/8) and (5/8) bucket. Yellow becomes (5/8+1=13/8) bucket, so the new ratio is (3/8:13/8=3:13).

Key conversions are (1\text{ m}=3.281\text{ ft}), (1\text{ m}^2=10.764\text{ ft}^2), (1\text{ acre}=43{,}560\text{ ft}^2), (1\text{ hectare}=10{,}000\text{ m}^2=2.471\text{ acres}), and (1\text{ L}=1000\text{ mL}). Temperature uses (F=\frac95C+32) and (C=\frac59(F-32)).

Divide both terms by 300: (600:900=2:3).

Each bus holds (162/3=54) people. Since (204/54=3\frac79), 4 buses are needed. They provide 216 seats, leaving 12 empty.

Delhi has about (30{,}000{,}000/1484\approx20{,}216) people/km². Mumbai has about (20{,}000{,}000/550\approx36{,}364) people/km². Mumbai is more crowded.

The neck is 4 of the 10 total ratio units, so its length would be (\\frac4{10}h=0.4h). Substitute your measured height for (h).

Using direct proportion, (x=\frac{(5/2)\times9}{3/7}=\frac52\times9\times\frac73=52.5) palas.

After (x) years, ((x+1):(x+5)=1:2). Thus (2x+2=x+5), so (x=3). Harmain will then be 4 and her brother 8.

(37:2::x:1), so (2x=37) and (x=18.5) kg.

The area is (100{,}000\text{ ft}^2). Using (1\text{ acre}=43{,}560\text{ ft}^2), manure needed is (10\times100{,}000/43{,}560\approx22.96) tonnes, or about 22,957 kg.

Ten litres is 10,000 mL, or 20 mugs. Time is (20\times15=300) seconds, which is 5 minutes.

Using (1\text{ acre}=43{,}560\text{ ft}^2), the cost is (₹15{,}00{,}000\times2400/43560\approx₹82{,}645).

Oxen take (20\times6=120) hours. A tractor is four times faster, so it takes (120/4=30) hours.

Copper mass is (\\frac34\times7.74=5.805) g and nickel mass is 1.935 g. Their costs are (5.805\times906/1000\approx₹5.26) and (1.935\times1341/1000\approx₹2.59), totaling about ₹7.85.

Puzzle 1 rows: HVHHVV, VHVVHH, VHHVHV, HVVHVH, HVHVHV, VHVHVH. Puzzle 2: HHVVHV, HVVHVH, VHHVHV, HVVHHV, VHHVVH, VVHHVH. Puzzle 3: HVHVHV, VHVHVH, HVHVVH, VHVHHV, HHVVHV, VVHHVH. Each row and column has three of each symbol, no run of three, and no duplicate row or column.