Chapter: Ratios and Proportions > Ratios and Proportions
Ratios and Proportions
Question 1 of 221 / 22
A proportion states that two ratios are equal. Step 1: Write the proportion as fractions. 2:5 = 8:x means 2/5 = 8/x Step 2: Cross multiply. 2 × x = 5 × 8 2x = 40 Step 3: Solve for x. x = 40 ÷ 2 x = 20 Check: 2:5 = 8:20 → 2/5 = 8/20 = 2/5 ✓ Therefore, x = 20.Definition: A proportion is an equation stating that two ratios are equal. If a:b = c:d, then a/b = c/d This means the ratio between the quantities stays the same. Example: 2:4 = 3:6 because both simplify to 1:2 The ratio 1:2 remains constant in both cases. Therefore, the statement is TRUE.Step 1: Find the cost per book. Cost per book = $15 ÷ 3 = $5 Step 2: Calculate the cost of 7 books. Cost of 7 books = 7 × $5 = $35 Alternative method using proportion: 3 books : $15 = 7 books : $x 3/15 = 7/x Cross multiply: 3x = 15 × 7 = 105 x = 105 ÷ 3 = 35 Therefore, 7 books would cost $35.Step 1: Find the speed (distance per hour). Speed = Distance ÷ Time Speed = 120 km ÷ 3 hours = 40 km/h Step 2: Calculate distance in 5 hours. Distance = Speed × Time Distance = 40 km/h × 5 hours = 200 km Alternative method using proportion: 120 km : 3 hours = x km : 5 hours 120/3 = x/5 x = (120 × 5) ÷ 3 = 600 ÷ 3 = 200 Therefore, the car would travel 200 km in 5 hours.Step 1: Find the cost per orange. Cost per orange = $10 ÷ 5 = $2 Step 2: Calculate the cost of 8 oranges. Cost of 8 oranges = 8 × $2 = $16 Alternative method using proportion: 5 oranges : $10 = 8 oranges : $x 5/10 = 8/x Cross multiply: 5x = 10 × 8 = 80 x = 80 ÷ 5 = 16 Therefore, 8 oranges would cost $16.Step 1: Find the flour per cookie. Flour per cookie = 2 cups ÷ 4 cookies = 0.5 cups per cookie Step 2: Calculate flour for 10 cookies. Flour for 10 cookies = 10 × 0.5 = 5 cups Alternative method using proportion: 2 cups : 4 cookies = x cups : 10 cookies 2/4 = x/10 Cross multiply: 4x = 2 × 10 = 20 x = 20 ÷ 4 = 5 Therefore, 5 cups of flour are needed for 10 cookies.Step 1: Identify the ratio. Ratio: 200 mL concentrate per 1 liter of juice Step 2: Set up the proportion. 200 mL : 1 liter = x mL : 5 liters Step 3: Calculate. x = 200 × 5 = 1000 mL Convert to liters: 1000 mL = 1 liter Therefore, 1000 mL (or 1 liter) of concentrate is needed for 5 liters of juice.This is an INVERSE proportion problem. More workers = less time, fewer workers = more time. Step 1: Find the total work (in worker-days). Total work = 4 workers × 6 days = 24 worker-days Step 2: Calculate time for 2 workers. Time = Total work ÷ Number of workers Time = 24 worker-days ÷ 2 workers = 12 days Check: 2 workers × 12 days = 24 worker-days ✓ Therefore, it would take 2 workers 12 days to build the wall.Step 1: Find the cost per kilogram. Cost per kg = $6 ÷ 2 kg = $3 per kg Step 2: Calculate the cost of 5 kg. Cost of 5 kg = 5 × $3 = $15 Alternative method using proportion: 2 kg : $6 = 5 kg : $x 2/6 = 5/x Cross multiply: 2x = 6 × 5 = 30 x = 30 ÷ 2 = 15 Therefore, 5 kg of apples would cost $15.Step 1: Find the scale (km per cm). Scale = 50 km ÷ 2 cm = 25 km per cm Step 2: Calculate the real distance for 5 cm. Real distance = 5 cm × 25 km/cm = 125 km Alternative method using proportion: 2 cm : 50 km = 5 cm : x km 2/50 = 5/x Cross multiply: 2x = 50 × 5 = 250 x = 250 ÷ 2 = 125 Therefore, 5 cm on the map represents 125 km in real distance.Step 1: Find the speed of the train. Speed = Distance ÷ Time Speed = 180 km ÷ 2 hours = 90 km/h Step 2: Calculate distance in 5 hours. Distance = Speed × Time Distance = 90 km/h × 5 hours = 450 km Alternative method using proportion: 180 km : 2 hours = x km : 5 hours 180/2 = x/5 x = (180 × 5) ÷ 2 = 900 ÷ 2 = 450 Therefore, the train will cover 450 km in 5 hours.Step 1: Find the weight per notebook. Weight per notebook = 300 g ÷ 6 = 50 g Step 2: Calculate the weight of 10 notebooks. Weight of 10 notebooks = 10 × 50 g = 500 g Alternative method using proportion: 6 notebooks : 300 g = 10 notebooks : x g 6/300 = 10/x Cross multiply: 6x = 300 × 10 = 3000 x = 3000 ÷ 6 = 500 Therefore, 10 notebooks would weigh 500 grams.Step 1: Find the paint per room. Paint per room = 4 liters ÷ 2 rooms = 2 liters per room Step 2: Calculate paint for 5 rooms. Paint for 5 rooms = 5 × 2 liters = 10 liters Alternative method using proportion: 4 liters : 2 rooms = x liters : 5 rooms 4/2 = x/5 Cross multiply: 2x = 4 × 5 = 20 x = 20 ÷ 2 = 10 Therefore, 10 liters of paint are needed for 5 rooms.Step 1: Convert time to the same unit. 45 minutes = 45/60 hours = 0.75 hours 2 hours = 120 minutes Step 2: Find the speed. Speed = 15 km ÷ 0.75 hours = 20 km/h Or: Speed = 15 km per 45 minutes = 1 km per 3 minutes Step 3: Calculate distance in 2 hours. Distance = 20 km/h × 2 hours = 40 km Or using minutes: 120 min ÷ 3 min × 1 km = 40 km Therefore, the cyclist can cover 40 km in 2 hours.Step 1: Find the sugar per cookie. Sugar per cookie = 3 cups ÷ 12 cookies = 0.25 cups per cookie Or as a fraction: 3/12 = 1/4 cup per cookie Step 2: Calculate sugar for 20 cookies. Sugar for 20 cookies = 20 × 0.25 = 5 cups Alternative method using proportion: 3 cups : 12 cookies = x cups : 20 cookies 3/12 = x/20 Cross multiply: 12x = 3 × 20 = 60 x = 60 ÷ 12 = 5 Therefore, 5 cups of sugar are needed for 20 cookies.Step 1: Find the rate (copies per minute). Rate = 100 copies ÷ 5 minutes = 20 copies per minute Step 2: Calculate copies in 12 minutes. Copies in 12 minutes = 20 × 12 = 240 copies Alternative method using proportion: 100 copies : 5 minutes = x copies : 12 minutes 100/5 = x/12 Cross multiply: 5x = 100 × 12 = 1200 x = 1200 ÷ 5 = 240 Therefore, the machine makes 240 copies in 12 minutes.Step 1: Find the output per carpenter (in 5 days). Chairs per carpenter = 15 chairs ÷ 3 carpenters = 5 chairs per carpenter Step 2: Calculate chairs made by 6 carpenters. Chairs by 6 carpenters = 6 × 5 = 30 chairs Alternative method using proportion: 3 carpenters : 15 chairs = 6 carpenters : x chairs 3/15 = 6/x Cross multiply: 3x = 15 × 6 = 90 x = 90 ÷ 3 = 30 Therefore, 6 carpenters can make 30 chairs in 5 days.Step 1: Identify the ratio. Ratio: 1 liter of water = 1 kilogram This is a 1:1 ratio between volume (in liters) and weight (in kg) for water. Step 2: Apply the ratio to 6 liters. 6 liters × 1 kg/liter = 6 kg Using proportion: 1 liter : 1 kg = 6 liters : x kg 1/1 = 6/x x = 6 Therefore, 6 liters of water would weigh 6 kilograms.Key concept: Distance = Speed × Time If two cyclists travel for the same time, their distances are proportional to their speeds. Speed ratio of cyclist A to cyclist B = 2:3 In the same time: Distance ratio of A to B = Speed ratio of A to B = 2:3 Examples of distances in ratio 2:3: • 20 km and 30 km • 40 km and 60 km • 8 km and 12 km • 100 km and 150 km Therefore, any pair of distances in the ratio 2:3 could be covered by the two cyclists in the same time.Step 1: Identify the scale. Scale: 1 cm on blueprint = 2 m in real life Step 2: Apply the scale to find actual length. Blueprint measurement = 6 cm Actual length = 6 cm × 2 m/cm = 12 m Using proportion: 1 cm : 2 m = 6 cm : x m 1/2 = 6/x Cross multiply: x = 2 × 6 = 12 Therefore, the actual length of the room is 12 meters.Step 1: Find the price per item. Price per item = $20 ÷ 5 items = $4 per item Step 2: Calculate the cost of 8 items. Cost of 8 items = 8 × $4 = $32 Alternative method using proportion: 5 items : $20 = 8 items : $x 5/20 = 8/x Cross multiply: 5x = 20 × 8 = 160 x = 160 ÷ 5 = 32 Therefore, 8 items would cost $32.Step 1: Identify the dosage ratio. Dosage ratio: 1 mL per 5 kg of body weight Step 2: Set up the proportion. 1 mL : 5 kg = x mL : 25 kg Step 3: Solve for x. 1/5 = x/25 Cross multiply: 5x = 1 × 25 = 25 x = 25 ÷ 5 = 5 Check: 25 kg ÷ 5 kg = 5, so 5 × 1 mL = 5 mL ✓ Therefore, a 25 kg child would receive 5 mL of medicine.
