Chapter 2.6 — Exercise 2.5 — Reducing Equations

Method 1 — Clearing Fractions with the LCM

When the variable itself is divided by different numbers, as in x/3 − x/4 = 14, the cleanest first move is to find the LCM of every denominator in the equation and multiply through by it. This clears all the fractions in a single step, leaving an ordinary equation you already know how to solve.

For x/3 − x/4 = 14: the LCM of 3 and 4 is 12, so multiplying through gives (4x − 3x)/12 = 14, which is x/12 = 14, so x = 168. With three denominators, as in z/2 + z/3 − z/6 = 8, the same idea applies to all three at once — the LCM of 2, 3 and 6 is 6, giving (3z + 2z − z)/6 = 8, or 4z/6 = 8, so z = 12.

When a mixed number appears, such as in 9¼ = y − 1⅓, convert it to an improper fraction (9¼ = 37/4, 1⅓ = 4/3) before finding any LCM — operating on the mixed form directly makes the LCM step far messier than it needs to be.

Method 2 — Cross-Multiplication

When the equation takes the form (expression)/(number) = (expression)/(number), cross-multiplication clears both denominators in one move: multiply the left numerator by the right denominator, and the right numerator by the left denominator.

For (2x − 3)/(3x + 2) = −2/3: cross-multiplying gives 3(2x − 3) = −2(3x + 2), which expands to 6x − 9 = −6x − 4, then 12x = 5, so x = 5/12. Once the cross-multiplication step is done, the rest is exactly the bracket-and-transposition process from Exercise 2.3.

Word Problems That Reduce to Fractional Equations

• A number defined by a fraction relationship: "the third part of a number exceeds its fifth part by 4" becomes x/3 − x/5 = 4; with LCM 15 this gives 2x/15 = 4, so x = 30.
• A fraction that changes with +1 on top and bottom: a fraction with numerator 4 less than its denominator becomes 1/2 once 1 is added to both parts. Writing the denominator as x, the fraction is (x−4)/x, and (x−3)/(x+1) = 1/2 gives x = 7, so the original fraction is 3/7.
• Splitting a total by a known fraction: in a class of 40 where girls are three-fifths of the number of boys, x + 3x/5 = 40 gives 8x/5 = 40, so there are 25 boys (and 15 girls).
• A total split three ways: a herd of deer where a quarter goes to the forest, a third grazes in a field, and the remaining 15 drink at a river gives x − x/4 − x/3 = 15; with LCM 12 this becomes 5x/12 = 15, so the herd has 36 deer.
• An age found from a future condition: if a person's age in 15 years will be four times their present age, x + 15 = 4x gives 3x = 15, so the present age is 5 years.
• Mixed coin denominations: a kiddy bank with three times as many 50-paise coins as one-rupee coins, worth ₹35 in total, gives x(1) + 3x(0.5) = 35, or 2.5x = 35, so there are 14 one-rupee coins and 42 fifty-paise coins.
• Work shared between two people: if two people together finish a task in 12 days, and one of them alone takes 20 days, the other's daily share is 1/12 − 1/20; with LCM 60 this is 2/60 = 1/x, so the second person alone would take 30 days.
• A time difference converted to hours: covering the same distance at 40 kmph instead of 50 kmph costs 6 extra minutes, i.e. 1/10 of an hour. Writing x/40 − x/50 = 1/10 and clearing with LCM 200 gives a distance of 20 km.
• A percentage gain on a sale: selling an item for ₹903 at a 5% gain means the selling price equals 105/100 of the cost price, so 21x/20 = 903 gives a cost price of ₹860.

Reading the Problem Before Reaching for LCM

Every one of these problems is solvable with ordinary transposition once the fractional equation is set up — the actual difficulty is almost always in translating the words correctly, not in the fraction arithmetic itself. "A quarter goes to the forest" is x/4, not x/4 of the remainder; "three times as many 50-paise coins as one-rupee coins" means the coin count is 3x, but its value in rupees is 3x × 0.5, not 3x itself. Reading each condition slowly enough to write down exactly what it says, before simplifying anything, avoids most of the errors that show up in this style of problem — the fraction arithmetic that follows is rarely where things go wrong — once the equation is written down correctly, the LCM or cross-multiplication steps are entirely mechanical from there.

Where the Fraction Techniques Break Down If Rushed

• Mixed numbers left unconverted. A value like 9¼ has to become 37/4 before it takes part in any LCM step — combining a whole number and a fraction directly under a common denominator produces a wrong result.
• An incomplete LCM. With three or more denominators in one equation, the LCM has to account for every one of them at once — taking the LCM of just two and treating the third separately gives an equation that doesn't clear all the fractions.
• Mismatched units in rate problems. Speed is given in kilometres per hour but a time difference is often given in minutes — convert minutes to hours (6 minutes = 6/60 = 1/10 hour) before the equation is written, not after.
• A negative numerator after cross-multiplication. When the right side of a cross-multiplied equation is negative, like −2/3, the negative sign applies to the whole resulting bracket, not just to one term inside it.

Closing Out Chapter 2's Equation-Solving Toolkit

The fraction-clearing and cross-multiplication techniques from this exercise reappear directly in Ratio and Proportion, where rates and shares are compared the same way, and cross-multiplication in particular becomes a standard tool again for rational equations in higher classes. For the bracket-expansion skill this exercise leans on once fractions are cleared, revisit Exercise 2.3. This closes out the equation-solving techniques of Chapter 2 — the same reasoning about forming and simplifying an equation carries forward into Algebraic Expressions next, where expressions get manipulated in their own right rather than always being set equal to something and solved.