Answer :
Sure, let's tackle each question one by one for a better understanding:
### Question 1: Simplifying Ratios
(a) 160:240
To simplify, we find the greatest common divisor (GCD) of 160 and 240. The GCD is 80.
- Simplified ratio: [tex]\( \frac{160}{80}:\frac{240}{80} = 2:3 \)[/tex]
(b) 12:15
Find the GCD of 12 and 15, which is 3.
- Simplified ratio: [tex]\( \frac{12}{3}:\frac{15}{3} = 4:5 \)[/tex]
(c) [tex]\(\frac{1}{2} : \frac{1}{8}\)[/tex]
Convert to the same denominator and simplify:
- Multiply both sides by 8 to clear fractions: [tex]\( \frac{1}{2} \times 8 : \frac{1}{8} \times 8 = 4:1 \)[/tex]
(e) [tex]\(\frac{1}{2}:1\)[/tex]
This is equivalent to [tex]\(\frac{1}{2}: \frac{2}{2}\)[/tex]. Simplify by multiplying both sides by 2:
- Simplified ratio: [tex]\(1:2\)[/tex]
(f) 0.4:0.2
Divide both numbers by 0.2 (GCD of 0.4 and 0.2):
- Simplified ratio: [tex]\( \frac{0.4}{0.2}:\frac{0.2}{0.2} = 2:1 \)[/tex]
(g) 27:0.3
Convert to whole numbers by multiplying both sides by 10:
- Simplified ratio: [tex]\(270:3 = 90:1\)[/tex]
(k) 0.006:0.3
Convert to whole numbers by multiplying both sides by 1000:
- Simplified ratio: [tex]\(6:300 = 1:50\)[/tex]
### Question 2: Simplifying Rates
(a) 1000 g in 6 glasses
Divide both by their GCD (2):
- Simplified rate: [tex]\( \frac{1000}{2}:\frac{6}{2} = 500:3 \)[/tex]
(b) M800 for 3 students
Divide by common factor (which is already simplified):
- Simplified rate: [tex]\( M800:3 \)[/tex]
(c) 80 km in 2 hours
Divide by 2:
- Simplified rate: [tex]\( 40:1 \)[/tex] (km per hour)
### Question 3: Write as a Unit Rate
(a) 828 Km in 6 days
Divide 828 by 6:
- Unit rate: [tex]\( 138.0 \)[/tex] Km per day
(b) M500 earned in 12 days
Divide 500 by 12:
- Unit rate: [tex]\( \approx 41.67 \)[/tex] M per day
(c) 42 cars for 3 countries
Divide 42 by 3:
- Unit rate: [tex]\( 14.0 \)[/tex] cars per country
### Question 4: Write in the form [tex]\(1: n\)[/tex]
(a) 2:5
To express 2 as 1, divide both sides by 2:
- Form: [tex]\(1:2.5\)[/tex]
(b) 8:40
Express 8 as 1, divide both sides by 8:
- Form: [tex]\(1:5\)[/tex]
### Question 5: Triangle Angle Ratios
Angles are in the ratio 3:2:4. Let the angles be [tex]\(3x\)[/tex], [tex]\(2x\)[/tex], and [tex]\(4x\)[/tex].
- Total [tex]\(3x + 2x + 4x = 180^\circ\)[/tex]
- [tex]\(9x = 180^\circ\)[/tex]
- [tex]\(x = 20^\circ\)[/tex]
Thus, angles are [tex]\(3x = 60^\circ\)[/tex], [tex]\(2x = 40^\circ\)[/tex], [tex]\(4x = 80^\circ\)[/tex].
### Question 6: Paint Mixture
Ratio is 4:7:3 and total is [tex]\(308 \, cm^3\)[/tex].
Total parts: [tex]\(4 + 7 + 3 = 14\)[/tex].
- Red: [tex]\( \frac{4}{14} \times 308 = 88.0 \, cm^3 \)[/tex]
- Blue: [tex]\( \frac{7}{14} \times 308 = 154.0 \, cm^3 \)[/tex]
- Yellow: [tex]\( \frac{3}{14} \times 308 = 66.0 \, cm^3 \)[/tex]
### Question 7: Building a Wall
3 men can build a wall in 10 hours. How many men for 5 hours?
Using the relation [tex]\[ \text{men} \times \text{hours} = \text{constant} \][/tex]
- [tex]\(3 \times 10 = x \times 5 \)[/tex]
- [tex]\(x = \frac{30}{5} = 6\)[/tex] men
### Question 8: Ploughing a Field
12 men can plough in 8 days. For 2 days:
- [tex]\(12 \times 8 = x \times 2\)[/tex]
- [tex]\(x = \frac{96}{2} = 48\)[/tex] men
### Question 9: Payment for Work Hours
Payment M376 for 8 hours. Find for 5 hours:
- [tex]\(\frac{M376}{8} \times 5 = M235.0\)[/tex]
### Question 10: Cost per Share
M12000 for 500 shares:
- [tex]\(\frac{M12000}{500} = M24.0\)[/tex] per share
### Question 11: Price Decrease Ratio
Old: M2500, New: M1600. Decrease is M900:
- Ratio of decrease to old price: [tex]\(900:2500\)[/tex]
### Question 12: Juice Ratios
Pineapple [tex]\(80 \, cm^3\)[/tex], Orange [tex]\(20 \, cm^3\)[/tex]:
- Pineapple to Orange: [tex]\(80:20\)[/tex]
- Orange to Total Mix: [tex]\(20:(80 + 20) = 20:100\)[/tex]
### Question 13: Ratio of Boys to Girls
12 boys and 18 girls:
- Ratio: [tex]\(12:18\)[/tex] which simplifies to [tex]\(2:3\)[/tex]
### Question 14: Passengers in Minibuses
64 passengers in 4 minibuses. Find how many in 6:
- Passengers per bus: [tex]\( \frac{64}{4} = 16\)[/tex]
- In 6 buses: [tex]\(16 \times 6 = 96\)[/tex]
### Question 15: Divide Weight in Ratio [tex]\(1:5:10\)[/tex]
Total weight: 320 kg.
- Total parts: [tex]\(1 + 5 + 10 = 16\)[/tex]
- First part: [tex]\( \frac{1}{16} \times 320 = 20 \, kg \)[/tex]
- Second part: [tex]\( \frac{5}{16} \times 320 = 100 \, kg \)[/tex]
- Third part: [tex]\( \frac{10}{16} \times 320 = 200 \, kg \)[/tex]
I hope this helps clarify each part of the question!
### Question 1: Simplifying Ratios
(a) 160:240
To simplify, we find the greatest common divisor (GCD) of 160 and 240. The GCD is 80.
- Simplified ratio: [tex]\( \frac{160}{80}:\frac{240}{80} = 2:3 \)[/tex]
(b) 12:15
Find the GCD of 12 and 15, which is 3.
- Simplified ratio: [tex]\( \frac{12}{3}:\frac{15}{3} = 4:5 \)[/tex]
(c) [tex]\(\frac{1}{2} : \frac{1}{8}\)[/tex]
Convert to the same denominator and simplify:
- Multiply both sides by 8 to clear fractions: [tex]\( \frac{1}{2} \times 8 : \frac{1}{8} \times 8 = 4:1 \)[/tex]
(e) [tex]\(\frac{1}{2}:1\)[/tex]
This is equivalent to [tex]\(\frac{1}{2}: \frac{2}{2}\)[/tex]. Simplify by multiplying both sides by 2:
- Simplified ratio: [tex]\(1:2\)[/tex]
(f) 0.4:0.2
Divide both numbers by 0.2 (GCD of 0.4 and 0.2):
- Simplified ratio: [tex]\( \frac{0.4}{0.2}:\frac{0.2}{0.2} = 2:1 \)[/tex]
(g) 27:0.3
Convert to whole numbers by multiplying both sides by 10:
- Simplified ratio: [tex]\(270:3 = 90:1\)[/tex]
(k) 0.006:0.3
Convert to whole numbers by multiplying both sides by 1000:
- Simplified ratio: [tex]\(6:300 = 1:50\)[/tex]
### Question 2: Simplifying Rates
(a) 1000 g in 6 glasses
Divide both by their GCD (2):
- Simplified rate: [tex]\( \frac{1000}{2}:\frac{6}{2} = 500:3 \)[/tex]
(b) M800 for 3 students
Divide by common factor (which is already simplified):
- Simplified rate: [tex]\( M800:3 \)[/tex]
(c) 80 km in 2 hours
Divide by 2:
- Simplified rate: [tex]\( 40:1 \)[/tex] (km per hour)
### Question 3: Write as a Unit Rate
(a) 828 Km in 6 days
Divide 828 by 6:
- Unit rate: [tex]\( 138.0 \)[/tex] Km per day
(b) M500 earned in 12 days
Divide 500 by 12:
- Unit rate: [tex]\( \approx 41.67 \)[/tex] M per day
(c) 42 cars for 3 countries
Divide 42 by 3:
- Unit rate: [tex]\( 14.0 \)[/tex] cars per country
### Question 4: Write in the form [tex]\(1: n\)[/tex]
(a) 2:5
To express 2 as 1, divide both sides by 2:
- Form: [tex]\(1:2.5\)[/tex]
(b) 8:40
Express 8 as 1, divide both sides by 8:
- Form: [tex]\(1:5\)[/tex]
### Question 5: Triangle Angle Ratios
Angles are in the ratio 3:2:4. Let the angles be [tex]\(3x\)[/tex], [tex]\(2x\)[/tex], and [tex]\(4x\)[/tex].
- Total [tex]\(3x + 2x + 4x = 180^\circ\)[/tex]
- [tex]\(9x = 180^\circ\)[/tex]
- [tex]\(x = 20^\circ\)[/tex]
Thus, angles are [tex]\(3x = 60^\circ\)[/tex], [tex]\(2x = 40^\circ\)[/tex], [tex]\(4x = 80^\circ\)[/tex].
### Question 6: Paint Mixture
Ratio is 4:7:3 and total is [tex]\(308 \, cm^3\)[/tex].
Total parts: [tex]\(4 + 7 + 3 = 14\)[/tex].
- Red: [tex]\( \frac{4}{14} \times 308 = 88.0 \, cm^3 \)[/tex]
- Blue: [tex]\( \frac{7}{14} \times 308 = 154.0 \, cm^3 \)[/tex]
- Yellow: [tex]\( \frac{3}{14} \times 308 = 66.0 \, cm^3 \)[/tex]
### Question 7: Building a Wall
3 men can build a wall in 10 hours. How many men for 5 hours?
Using the relation [tex]\[ \text{men} \times \text{hours} = \text{constant} \][/tex]
- [tex]\(3 \times 10 = x \times 5 \)[/tex]
- [tex]\(x = \frac{30}{5} = 6\)[/tex] men
### Question 8: Ploughing a Field
12 men can plough in 8 days. For 2 days:
- [tex]\(12 \times 8 = x \times 2\)[/tex]
- [tex]\(x = \frac{96}{2} = 48\)[/tex] men
### Question 9: Payment for Work Hours
Payment M376 for 8 hours. Find for 5 hours:
- [tex]\(\frac{M376}{8} \times 5 = M235.0\)[/tex]
### Question 10: Cost per Share
M12000 for 500 shares:
- [tex]\(\frac{M12000}{500} = M24.0\)[/tex] per share
### Question 11: Price Decrease Ratio
Old: M2500, New: M1600. Decrease is M900:
- Ratio of decrease to old price: [tex]\(900:2500\)[/tex]
### Question 12: Juice Ratios
Pineapple [tex]\(80 \, cm^3\)[/tex], Orange [tex]\(20 \, cm^3\)[/tex]:
- Pineapple to Orange: [tex]\(80:20\)[/tex]
- Orange to Total Mix: [tex]\(20:(80 + 20) = 20:100\)[/tex]
### Question 13: Ratio of Boys to Girls
12 boys and 18 girls:
- Ratio: [tex]\(12:18\)[/tex] which simplifies to [tex]\(2:3\)[/tex]
### Question 14: Passengers in Minibuses
64 passengers in 4 minibuses. Find how many in 6:
- Passengers per bus: [tex]\( \frac{64}{4} = 16\)[/tex]
- In 6 buses: [tex]\(16 \times 6 = 96\)[/tex]
### Question 15: Divide Weight in Ratio [tex]\(1:5:10\)[/tex]
Total weight: 320 kg.
- Total parts: [tex]\(1 + 5 + 10 = 16\)[/tex]
- First part: [tex]\( \frac{1}{16} \times 320 = 20 \, kg \)[/tex]
- Second part: [tex]\( \frac{5}{16} \times 320 = 100 \, kg \)[/tex]
- Third part: [tex]\( \frac{10}{16} \times 320 = 200 \, kg \)[/tex]
I hope this helps clarify each part of the question!
