Scaling Techniques: – Adjusting recipes and solving other multiplicative. comparisons.

SCALING TECHNIQUES Adjusting recipes and solving other multiplicative. comparisons. Grade 7 Mathematics: Scaling Techniques – Adjusting Recipes and Solving Multiplicative Comparisons Subtopic Navigator Introduction Understanding Scaling and Ratios Scaling Up Recipes Scaling Down Recipes Multiplicative Comparisons Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Understand the concept of scaling using ratios and proportions. Learn how to scale up recipes when cooking for more people. Learn how to scale down recipes when cooking for fewer people. Apply scaling techniques to solve real-life multiplicative comparison problems. Lesson Introduction Scaling techniques involve adjusting quantities in proportion to one another. For example, when a recipe designed for 4 people must be adjusted to serve 10, all ingredients must be increased in the same ratio. This is achieved using the mathematical ideas of ratios and proportions. Scaling can be up (increasing) or down (decreasing). Multiplicative comparisons also use scaling, comparing one quantity as a multiple of another. For instance, if a car is 3 times as fast as a bicycle, then the ratio is [latex]3:1[/latex]. Understanding Scaling and Ratios Scaling is based on the principle of proportionality: if all parts of a situation are multiplied by the same factor, the relationships remain the same. Suppose a recipe uses [latex]2[/latex] cups of flour and [latex]3[/latex] cups of milk. The ratio is [latex]2:3[/latex]. Doubling both gives [latex]4:6[/latex], which is still the same proportion. Example 1: A juice recipe calls for [latex]2[/latex] cups of water and [latex]1[/latex] cup of concentrate. If the ratio is maintained, how much concentrate is needed for [latex]6[/latex] cups of water? Solution: The ratio is [latex]2:1[/latex]. For [latex]6[/latex] cups water: [latex]dfrac{6}{2} = 3[/latex]. So, [latex]3 times 1 = 3[/latex] cups concentrate. Example 2: A scale drawing uses [latex]1[/latex] cm to represent [latex]5[/latex] m. How many meters are represented by [latex]12[/latex] cm? Solution: [latex]12 times 5 = 60[/latex]. So, it represents [latex]60[/latex] meters. Example 3: A chemical mixture is [latex]4[/latex] parts water to [latex]1[/latex] part salt. If [latex]20[/latex] parts of water are used, how much salt is required? Solution: [latex]dfrac{20}{4} = 5[/latex]. So, [latex]5 times 1 = 5[/latex] parts salt. Exercises (Understanding Scaling) A toy car is scaled so that [latex]1[/latex] cm represents [latex]10[/latex] cm in real life. How long is a toy car representing a [latex]4.5[/latex] m real car? A recipe is in the ratio sugar : butter = [latex]3:2[/latex]. If [latex]9[/latex] cups of sugar are used, how much butter is needed? Scaling Up Recipes Scaling up means increasing quantities while keeping proportions the same. This is common when cooking for more people. Multiply each ingredient by the scale factor. Example 4: A cake recipe for 5 people requires 200 g of flour. How much flour is needed for 20 people? Solution: Scale factor = [latex]dfrac{20}{5} = 4[/latex]. Required flour = [latex]200 times 4 = 800[/latex] g. Example 5: A recipe for 2 people needs 1.5 liters of juice. How much is needed for 10 people? Solution: Scale factor = [latex]dfrac{10}{2} = 5[/latex]. Required = [latex]1.5 times 5 = 7.5[/latex] liters. Example 6: A salad recipe for 4 people requires 3 tomatoes. How many tomatoes are needed for 14 people? Solution: Scale factor = [latex]dfrac{14}{4} = 3.5[/latex]. Required = [latex]3 times 3.5 = 10.5[/latex] tomatoes (≈ 11 tomatoes). Exercises (Scaling Up) A recipe for 8 people uses 600 g of rice. How much rice is needed for 20 people? A juice recipe for 3 people uses 2 lemons. How many lemons are needed for 18 people? Scaling Down Recipes Scaling down means reducing all ingredients in the same proportion. This is used when cooking for fewer people. Example 7: A stew recipe for 12 people uses 1.2 kg of meat. How much meat is needed for 4 people? Solution: Scale factor = [latex]dfrac{4}{12} = dfrac{1}{3}[/latex]. Required meat = [latex]1.2 times dfrac{1}{3} = 0.4[/latex] kg. Example 8: A juice recipe for 10 people uses 5 liters of water. How much for 2 people? Solution: Scale factor = [latex]dfrac{2}{10} = 0.2[/latex]. Required water = [latex]5 times 0.2 = 1[/latex] liter. Example 9: A soup recipe for 15 people uses 600 g of carrots. How much for 5 people? Solution: Scale factor = [latex]dfrac{5}{15} = dfrac{1}{3}[/latex]. Required carrots = [latex]600 times dfrac{1}{3} = 200[/latex] g. Exercises (Scaling Down) A meal recipe for 20 people uses 5 kg of yam. How much is needed for 8 people? A salad recipe for 12 people uses 24 onions. How many onions for 3 people? Multiplicative Comparisons Multiplicative comparison problems compare one quantity as a multiple of another. For example: "John is 3 times older than Peter." This is solved by multiplying or dividing by the factor. Example 10: A rope is 4 times as long as another. If the shorter is 3 m, how long is the longer? Solution: [latex]3 times 4 = 12[/latex] m. Example 11: A pen costs ₦50. A bag costs 10 times as much. How much is the bag? Solution: [latex]50 times 10 = ₦500[/latex]. Example 12: Musa is 5 times older than his sister. If his sister is 6 years, how old is Musa? Solution: [latex]6 times 5 = 30[/latex] years. Exercises (Multiplicative Comparisons) A car is 8 times as fast as a bicycle. If the bicycle moves at 15 km/h, how fast is the car? A book costs ₦250. A calculator costs 12 times as much. How much is the calculator? Applications and Mixed Problems Example 13: A family recipe for 6 people uses 900 g of chicken. The family is hosting 15 people. How much chicken is needed? Solution: Scale factor = [latex]dfrac{15}{6} = 2.5[/latex]. Required chicken = [latex]900 times 2.5 = 2250[/latex] g. Example 14: A building plan shows 1 cm representing 2 m. How many meters does 8.5 cm represent? Solution: [latex]8.5 times 2 = 17[/latex] m. Example 15: A fruit punch recipe for 5 people uses 2.5 liters. How much is needed for 18 people? Solution: Scale factor = [latex]dfrac{18}{5} = 3.6[/latex]. Required = [latex]2.5 times 3.6 = 9[/latex] liters. Exercises (Applications) A recipe for 4 people uses 2 cups of rice. How much for 14 people? A car uses 12 liters of petrol for 180 km. How much petrol is needed for 450 km? Cumulative Exercises A toy is built to scale so that 1 cm represents 20 cm. If the toy is 15 cm, what is the actual length? A recipe for 10 people uses 5 liters of water. How much for 25 people? A bag is 6 times heavier than a book. If the book is 2 kg, what is the weight of the bag? A juice recipe for 8 people uses 4 oranges. How many for 20 people? A car covers 150 km in 3 hours. At the same speed, how far in 5 hours? A recipe for 12 people uses 24 cups of milk. How much for 3 people? A man is 5 times taller than a boy. If the boy is 1.2 m, how tall is the man? A scale drawing shows 1 cm = 50 m. What distance does 7.5 cm represent? A party recipe for 5 people requires 10 eggs. How many for 18 people? A bus fare is ₦200. A plane ticket costs 25 times as much. How much is the plane ticket? Conclusion/Recap Scaling techniques help us adjust quantities while keeping proportions constant. Whether scaling up or down, or making multiplicative comparisons, the key is to multiply or divide by the same factor. This skill is useful in real-life problems such as cooking, construction, and speed-distance-time situations. In exams, such questions usually test ratio, proportion, and logical application of scaling. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c