Simple Arithmetic Equations — Study Notes

Overview

Simple arithmetic equations form the backbone of problem-solving in the UPSSSC PET Elementary Arithmetic section. These questions test your ability to translate real-world scenarios into mathematical expressions and solve for unknown values. Typically, you'll encounter 3–5 direct questions on linear equations, plus several word problems that require equation-building skills.

Mastering this topic is crucial because it's not just a standalone section—equation-solving skills underpin percentage problems, age-related questions, time-distance scenarios, and mixture problems. The exam consistently features straightforward single-variable linear equations alongside word problems involving money, ages, consecutive numbers, and basic transactions. Students who can quickly identify the variable, set up the equation correctly, and solve systematically gain valuable marks and save time for tougher questions.

The key challenge isn't the mathematics itself—it's translating Hindi/English problem statements into correct equations and avoiding calculation errors under time pressure. Focus on recognizing common problem patterns and developing a systematic solving approach.

Key Concepts

• **Linear equation in one variable**: An equation of the form ax + b = c, where a, b, c are known numbers and x is the unknown. The highest power of the variable is 1.

• **Solving principle**: Whatever operation you perform on one side of the equation, perform the same on the other side to maintain equality. This includes adding, subtracting, multiplying, or dividing both sides by the same non-zero number.

• **Transposition method**: Moving terms from one side to the other by changing their sign. If +5 is on the left, it becomes –5 when moved to the right; if ×3 is with x, it becomes ÷3 when moved across the equals sign.

• **Word problem translation**: "Sum" means addition (+), "difference" means subtraction (–), "product" means multiplication (×), "quotient" means division (÷), "is/was/will be" translates to equals (=).

• **Consecutive numbers**: If first number is x, then next consecutive integers are x+1, x+2, etc. For consecutive even numbers: x, x+2, x+4. For consecutive odd numbers: x, x+2, x+4.

• **Age problems pattern**: If present age is x, then age after n years = x+n, and age n years ago = x–n.

• **Verification step**: Always substitute your answer back into the original equation or problem statement to check correctness—this catches arithmetic errors.

• **Unknown identification**: In word problems, identify what you need to find and assign it as your variable (x). Express all other quantities in terms of this variable.

Formulas / Key Facts

• **Basic linear equation**: ax + b = c → Solution: x = (c – b)/a

• **Equation with variable on both sides**: ax + b = cx + d → Collect like terms → (a – c)x = d – b → x = (d – b)/(a – c)

• **Equation with fractions**: x/a + x/b = c → Take LCM and clear denominators first → bx + ax = abc → x(b + a) = abc → x = abc/(a + b)

• **Three consecutive integers sum**: x + (x+1) + (x+2) = given sum → 3x + 3 = sum

• **Two-part division problems**: If total is T and one part is x, the other part is (T – x)

• **Price-quantity relationship**: Total cost = Price per unit × Number of units

• **Work distribution**: If total work is W and x is done by first person, then (W – x) remains for others

• **Percentage to equation**: "x is 20% more than y" translates to x = y + 0.20y = 1.20y

Worked Examples

**Example 1: Basic linear equation** Solve: 3x – 7 = 20

*Solution:* Step 1: Add 7 to both sides: 3x – 7 + 7 = 20 + 7 Step 2: Simplify: 3x = 27 Step 3: Divide both sides by 3: x = 27/3 = 9 *Verification:* 3(9) – 7 = 27 – 7 = 20 ✓

**Example 2: Variable on both sides** Solve: 5x + 8 = 2x + 26

*Solution:* Step 1: Bring x-terms to left: 5x – 2x + 8 = 26 Step 2: Simplify: 3x + 8 = 26 Step 3: Subtract 8: 3x = 18 Step 4: Divide by 3: x = 6 *Verification:* 5(6) + 8 = 38 and 2(6) + 26 = 38 ✓

**Example 3: Word problem—consecutive numbers** The sum of three consecutive odd numbers is 63. Find the numbers.

*Solution:* Step 1: Let first odd number = x Step 2: Next two consecutive odd numbers = x+2 and x+4 Step 3: Set up equation: x + (x+2) + (x+4) = 63 Step 4: Simplify: 3x + 6 = 63 Step 5: Subtract 6: 3x = 57 Step 6: Divide by 3: x = 19 *Answer:* The three numbers are 19, 21, and 23. *Verification:* 19 + 21 + 23 = 63 ✓

**Example 4: Money distribution problem** Ravi has ₹500 in notes of ₹50 and ₹100. If he has 7 notes in total, how many ₹100 notes does he have?

*Solution:* Step 1: Let number of ₹100 notes = x Step 2: Then number of ₹50 notes = 7 – x Step 3: Total value equation: 100x + 50(7 – x) = 500 Step 4: Expand: 100x + 350 – 50x = 500 Step 5: Simplify: 50x + 350 = 500 Step 6: Subtract 350: 50x = 150 Step 7: Divide by 50: x = 3 *Answer:* 3 notes of ₹100 and 4 notes of ₹50.

Common Mistakes

**Mistake 1**: Sign errors when transposing → *Correct fix*: Remember that +b becomes –b and ×a becomes ÷a when crossing the equals sign. Write each step clearly rather than doing mental math.

**Mistake 2**: Forgetting to apply operations to both sides → *Correct fix*: Whatever you do to one side (add 5, multiply by 2, etc.), you MUST do to the other side. Treat the equals sign as a balance scale.

**Mistake 3**: Incorrect translation of "more than" and "less than" → *Correct fix*: "5 more than x" is x+5, not 5+x (though algebraically same). "5 less than x" is x–5, NOT 5–x (these are different!).

**Mistake 4**: Setting up wrong relationships in word problems → *Correct fix*: Read the problem twice. Underline what you need to find. Clearly define your variable before writing the equation. If problem says "total is 100," your equation must equal 100.

**Mistake 5**: Calculation errors with negative numbers → *Correct fix*: When solving 2x = –10, remember x = –5, not +5. When –3x = 12, then x = 12/(–3) = –4. Pay attention to signs during division.

Quick Reference

• To solve ax + b = c: Isolate x by doing inverse operations in reverse order of BODMAS.

• Word problem strategy: Read → Define variable → Translate to equation → Solve → Verify answer makes sense.

• Consecutive integers: x, x+1, x+2. Consecutive even/odd: x, x+2, x+4 (same pattern).

• Always verify: Substitute your answer back into original equation to catch errors.

• Fraction equations: First step is to clear denominators by multiplying through by LCM.

• "Sum of" = add all parts. "Difference between" = subtract smaller from larger.