Exercise 2.5 — Reducing Equations

Solving Linear Equations Involving Fractions

Exercise 2.5 is the most varied and demanding exercise in Chapter 2 for Class 8 students. It covers two major skill areas: solving equations where the variable terms appear as fractions (requiring LCM), and solving equations where an entire expression sits over a denominator (requiring cross-multiplication). These question types are consistently tested in CBSE, Telangana, and Andhra Pradesh board examinations, and mastering them is essential before moving into higher algebra.

Method 1 — Equations with Fractional Coefficients (Using LCM)

When the variable itself is divided by different numbers on both sides — for example x/3 − x/4 = 14 — the cleanest approach is to find the LCM of all denominators and multiply every term through. This clears all fractions in one step and converts the equation into a simple form you can solve by transposition.

The process: find the LCM, rewrite each fraction with the common denominator, combine like terms on each side, then isolate the variable. For x/3 − x/4 = 14, the LCM of 3 and 4 is 12, giving (4x − 3x)/12 = 14, so x/12 = 14, and therefore x = 168.

When three fractional terms appear — as in z/2 + z/3 − z/6 = 8 — find the LCM of all three denominators (here, LCM of 2, 3, and 6 is 6), combine over the common denominator, and solve. Problems with mixed numbers (like Q(v): 9¼ = y − 1⅓) require converting mixed numbers to improper fractions first, then proceeding with LCM as usual.

Method 2 — Equations with Expressions over Denominators (Cross-Multiplication)

When the equation has the form (expression) / (number) = (expression) / (number), the most efficient method is cross-multiplication: multiply the numerator on the left by the denominator on the right, and vice versa. This eliminates both denominators at once.

For example, (2x − 3)/(3x + 2) = −2/3 becomes 3(2x − 3) = −2(3x + 2), which expands to 6x − 9 = −6x − 4, then 12x = 5, giving x = 5/12. Notice that after cross-multiplying, the problem reduces to a standard brackets-and-transposition problem from Exercise 2.3.

Word Problems Involving Fractions

The second half of this exercise applies both methods to real-life scenarios, combining the equation-forming skill from Exercise 2.4 with the fraction-solving technique introduced here.

• Fraction of a total (Q2, Q6, Q11) — "The third part of a number exceeds its fifth part by 4" gives x/3 − x/5 = 4; LCM of 3 and 5 is 15, leading to x = 30. Similarly, finding the number of boys in a class when girls are three-fifths of boys, or counting deer split between forest, field, and river — all use the same fraction-of-a-whole setup.
• Work and time (Q9) — A key problem type: if A and B together complete work in 12 days and A alone takes 20 days, the part B does per day is 1/12 − 1/20. Using LCM 60 gives 2/60 = 1/x, so x = 30 days.
• Speed, distance and time (Q10) — The difference in time between running at 40 kmph and 50 kmph is 6 minutes (= 6/60 hr). Setting x/40 − x/50 = 1/10 and using LCM 200 gives x = 20 km. Remember: always convert minutes to hours before forming the equation.
• Profit and selling price (Q12) — If a shopkeeper sells a radio for ₹903 at a 5% gain, the selling price equals x × 105/100 = 21x/20 = 903, so the cost price is ₹860. This links linear equations directly to the Comparing Quantities chapter.
• Coins and currency (Q8) — Mixing rupee values of different coin types: let one-rupee coins = x and 50-paise coins = 3x. The total value equation becomes x + 3x(1/2) = 35, giving x = 14 one-rupee coins and 42 fifty-paise coins.

Common Mistakes to Avoid

• Forgetting to convert mixed numbers — Always change mixed numbers like 9¼ to improper fractions (37/4) before applying LCM. Working with mixed numbers directly leads to errors.
• Using the wrong LCM — In multi-denominator problems, find the LCM of all denominators at once. A common error is taking LCM of just two denominators and forgetting the third.
• Units mismatch in speed-time problems — Speed is in kmph but time differences are given in minutes. Always convert: 6 minutes = 6/60 = 1/10 hours before writing the equation.
• Sign errors after cross-multiplication — When the RHS has a negative numerator (e.g. −2/3), the negative sign must be applied to the entire bracket after cross-multiplication.

What This Exercise Prepares You For

The fraction-solving and cross-multiplication techniques from this exercise are foundational for later chapters. Work-time and speed-distance problems reappear in Comparing Quantities and in Class 9 and 10 problem sets. The cross-multiplication method is directly reused when solving rational equations in Class 10. Students who also want to revisit the bracket-expansion techniques needed after cross-multiplication can refer back to Exercise 2.3.