What is Mixture Problems using Algebra?

S3-SA1-0418

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Mixture problems using algebra involve combining two or more things (like liquids, ingredients, or items) with different properties to create a new mixture. We use algebraic equations to find unknown quantities, such as the amount of each ingredient or the final concentration of the mixture.

Simple Example

Quick Example

Imagine you have two types of chai powder: one strong and one mild. If you mix 100 grams of the strong powder with 200 grams of the mild powder, you create a new mixture. A mixture problem would ask you to find the percentage of strong powder in your final mix.

Worked Example

Step-by-Step

PROBLEM: A shopkeeper mixes two varieties of rice. Variety A costs Rs 60 per kg, and Variety B costs Rs 80 per kg. He wants to make a 20 kg mixture that costs Rs 65 per kg. How many kgs of each variety should he use?

Step 1: Define variables.
Let 'x' be the amount (in kg) of Variety A rice.
Since the total mixture is 20 kg, the amount of Variety B rice will be (20 - x) kg.

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Step 2: Write an equation for the total cost.
The total cost of Variety A rice is 60 * x.
The total cost of Variety B rice is 80 * (20 - x).
The total cost of the mixture is 65 * 20.
So, 60x + 80(20 - x) = 65 * 20.

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Step 3: Simplify and solve the equation.
60x + 1600 - 80x = 1300

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Step 4: Combine like terms.
-20x + 1600 = 1300

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Step 5: Isolate the variable.
-20x = 1300 - 1600
-20x = -300

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Step 6: Solve for x.
x = -300 / -20
x = 15

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Step 7: Find the amount of Variety B.
Variety A = x = 15 kg.
Variety B = 20 - x = 20 - 15 = 5 kg.

Answer: The shopkeeper should use 15 kg of Variety A rice and 5 kg of Variety B rice.

Why It Matters

Mixture problems are super useful in many fields! Data scientists use them to combine different datasets, and chemical engineers use them to mix solutions for new products. Even in finance, they help calculate average returns from different investments, making careers like financial analyst or research scientist possible.

Common Mistakes

MISTAKE: Students often forget to account for the total quantity or total value of the final mixture. | CORRECTION: Always ensure your equation balances the total amount of the ingredients with the total amount of the mixture, and similarly for their values or concentrations.

MISTAKE: Incorrectly setting up the variable for the second component. For example, if one component is 'x' and total is 'T', they might use 'y' for the second instead of 'T-x'. | CORRECTION: If you define one unknown as 'x' and you know the total, express the second unknown in terms of 'x' and the total (e.g., total - x). This reduces the number of variables to solve for.

MISTAKE: Making calculation errors when distributing numbers in an equation, like 80(20-x) becoming 80*20 - x instead of 80*20 - 80*x. | CORRECTION: Always apply the distributive property carefully, multiplying the outside number by EACH term inside the parentheses.

Practice Questions

Try It Yourself

QUESTION: A juice vendor mixes 5 liters of orange juice (costing Rs 40/liter) with 3 liters of apple juice (costing Rs 60/liter). What is the cost per liter of the mixture? | ANSWER: Rs 47.50 per liter

QUESTION: A chemist has two solutions of acid: one is 30% acid and the other is 70% acid. How many milliliters of each solution should she use to make 100 ml of a 40% acid solution? | ANSWER: 75 ml of 30% acid solution and 25 ml of 70% acid solution

QUESTION: A sweet shop owner wants to make 10 kg of a new laddoo mix. He has besan laddoos costing Rs 300/kg and motichoor laddoos costing Rs 450/kg. If he wants the mixture to sell for Rs 360/kg, how many kgs of motichoor laddoos should he use? | ANSWER: 4 kg of motichoor laddoos

MCQ

Quick Quiz

You mix 2 kg of sugar costing Rs 50/kg with 3 kg of sugar costing Rs 60/kg. What is the total cost of the mixture?

Rs 100

Rs 180

Rs 280

Rs 300

The Correct Answer Is:

The cost of the first sugar is 2 kg * Rs 50/kg = Rs 100. The cost of the second sugar is 3 kg * Rs 60/kg = Rs 180. The total cost is Rs 100 + Rs 180 = Rs 280.

Real World Connection

In the Real World

Mixture problems are used by food scientists to create new recipes, ensuring the right balance of ingredients for taste and nutrition. They're also used in pharmacies to mix medicines with precise concentrations, or by petrol pump owners to blend different grades of fuel for optimal performance.

Key Vocabulary

Key Terms

MIXTURE: A substance made by combining two or more different ingredients | ALGEBRAIC EQUATION: A mathematical statement showing two expressions are equal, used to find unknown values | CONCENTRATION: The amount of one substance in a mixture, often expressed as a percentage | VARIABLE: A symbol (like 'x') used to represent an unknown number in an equation

What's Next

What to Learn Next

Great job understanding mixture problems! Next, you can explore 'Ratio and Proportion' or 'Percentage Applications'. These concepts will help you solve even more complex real-world problems involving quantities and their relationships.