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Applying ratio and rate concepts to solve everyday math challenges using diagrams and logic.
Imagine you are a world-class chef preparing a secret sauce for 100 people, but your recipe only serves 4. If you guess the ingredients, the taste will be ruined—how do you use math to keep the flavor perfectly balanced?
A ratio is a way to compare two or more quantities. When word problems get tricky, we use a tape diagram to see the math. Think of a tape diagram as a series of equal-sized boxes. Each box represents the same 'value' or amount. If the ratio of blue paint to yellow paint is , you draw 2 boxes for blue and 3 boxes for yellow. Because every box is identical, if you find the value of one box, you've unlocked the whole problem!
A painter mixes blue and red paint in a ratio of . If she uses 10 gallons of blue paint, how much red paint does she need?
1. Draw 2 boxes for Blue and 3 boxes for Red. 2. We know the 2 Blue boxes represent 10 gallons. 3. Divide the total blue by the number of boxes: . Each box is worth 5 gallons. 4. Since every box is equal, the 3 Red boxes are also worth 5 each: . 5. Answer: She needs 15 gallons of red paint.
Quick Check
If a ratio of apples to oranges is , how many total boxes would you draw in your tape diagram?
Answer
5 boxes (4 for apples + 1 for oranges).
Sometimes we don't need a diagram; we just need a multiplier. This is called scaling. If you have a ratio of , you can create an equivalent ratio by multiplying both sides by the same number, . For example, if is your base, multiplying by 5 gives you . This 'k' is your scale factor. It tells you how many times bigger or smaller your actual amounts are compared to the starting ratio.
The ratio of students to chaperones is . If there are 120 students going on the trip, how many chaperones are needed?
1. Set up the ratio: . 2. We know the new student count is 120. Find the scale factor: . 3. Multiply the chaperone side by the same scale factor: . 4. Answer: 15 chaperones are needed.
Quick Check
If you scale the ratio by a factor of 4, what is the new ratio?
Answer
The most common challenge is when a problem gives you the total amount of everything combined. To solve these, you first find the sum of the ratio parts. This tells you how many 'units' exist in the whole group. Then, divide the total quantity by the total units to find the value of a single unit. Once you know what one unit is worth, you can find the value of any specific part.
A construction worker mixes cement, sand, and gravel in a ratio of . The total weight of the mixture is 600 pounds. How many pounds of sand are in the mix?
1. Find the total parts: parts. 2. Find the value of one part: pounds per part. 3. The ratio for sand is 2 parts. 4. Multiply the sand parts by the value per part: . 5. Answer: There are 200 pounds of sand.
A recipe uses a ratio of flour to sugar. If you use 6 cups of sugar, how much flour do you need?
In a bag of 40 marbles, the ratio of red to blue is . How many red marbles are there?
In a tape diagram, the boxes for the first part of the ratio can be larger than the boxes for the second part.
Review Tomorrow
In 24 hours, try to explain to a friend how to use a tape diagram to solve a ratio problem where the total is 70.
Practice Activity
Go to your kitchen and find a food label with a serving size. Calculate the ratio of sugar to total carbohydrates, then figure out how much sugar you would eat if you had 5 servings.