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Simple Equations - Applications of Simple Equations to practical situations

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Understanding the Variable: In practical situations, we first identify the unknown quantity and represent it with a letter like x,y,z,n,x, y, z, n, or pp. This variable acts as a placeholder for the numerical value we are trying to find.

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Translating Statements to Expressions: Word problems use specific keywords that translate to mathematical operations. For example, 'sum' or 'increased by' indicates addition (++), while 'difference' or 'less than' indicates subtraction (−-). 'Times' or 'product' indicates multiplication (×\times), and 'ratio' or 'divided by' indicates division (÷\div).

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Setting up the Equation: To form an equation, we represent the relationship between the variable and given numbers. Imagine a balance scale where the 'is equal to' sign (==) acts as the central pivot. The total value on the left-hand side (LHS) must perfectly balance the total value on the right-hand side (RHS).

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Solving the Equation using Inverse Operations: To find the value of the variable, we 'undo' the operations applied to it. If a number is added, we subtract it from both sides; if a number is multiplied, we divide both sides by it. This process keeps the equation's 'balance' intact until the variable is isolated on one side.

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Number and Age Problems: Practical applications often involve finding a person's age or a specific number. In age problems, if a person's current age is xx, their age nn years ago was x−nx - n, and their age nn years from now will be x+nx + n. Visualizing a timeline can help organize these different time periods.

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Geometry-based Applications: Many equations relate to geometric shapes. For instance, finding the length of a side given the perimeter. Imagine a rectangle where the perimeter is the total boundary length; the equation is formed by adding all four sides: 2(l+w)=P2(l + w) = P.

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Verifying the Result: After finding the numerical value of the variable, it is crucial to substitute it back into the original word problem. The value must make logical sense in the context (e.g., age cannot be negative) and satisfy the original balance of the equation.

📐Formulae

General form of a simple equation: ax+b=cax + b = c

Sum of numbers: x+y=Sx + y = S

Difference of numbers: x−y=Dx - y = D

Product of numbers: ax=Pax = P

Consecutive integers: x,(x+1),(x+2),...x, (x+1), (x+2), ...

Perimeter of a Rectangle: P=2(l+b)P = 2(l + b)

Relationship for age: Future Age=Current Age+n\text{Future Age} = \text{Current Age} + n

💡Examples

Problem 1:

The sum of three times a number and 11 is 32. Find the number.

Solution:

Step 1: Let the unknown number be xx. Step 2: According to the problem, three times the number is 3x3x. Adding 11 to it gives 3x+113x + 11. Step 3: Set up the equation: 3x+11=323x + 11 = 32. Step 4: Subtract 11 from both sides: 3x=32−113x = 32 - 11, which gives 3x=213x = 21. Step 5: Divide both sides by 3: x=213x = \frac{21}{3}. Step 6: Therefore, x=7x = 7.

Explanation:

We first translated the phrase 'three times a number and 11' into the algebraic expression 3x+113x + 11. We then used the balancing method to isolate xx by performing inverse operations (subtraction then division).

Problem 2:

Rohan's father is 49 years old. He is 4 years older than three times Rohan's age. What is Rohan's age?

Solution:

Step 1: Let Rohan's age be yy years. Step 2: Three times Rohan's age is 3y3y. Step 3: The father is 4 years older than 3y3y, so the father's age is 3y+43y + 4. Step 4: We are given the father's age is 49, so 3y+4=493y + 4 = 49. Step 5: Transpose 4 to the RHS: 3y=49−43y = 49 - 4, so 3y=453y = 45. Step 6: Divide by 3: y=453y = \frac{45}{3}. Step 7: y=15y = 15. Rohan is 15 years old.

Explanation:

In this age problem, we defined the unknown (Rohan's age) as yy. We then constructed the father's age relative to yy and set it equal to the known constant (49) to solve for yy.