Ratio and Proportion, Indices and Logarithms for CA Foundation

What this chapter covers

Ratio and Proportion, Indices and Logarithms: topics in the order to study them

1. 1Ratio: Meaning, Types and PropertiesEverything else in the first half builds on what a ratio is and how to compare and simplify ratios.
2. 2Proportion and Continued ProportionProportion is two equal ratios, so you need ratio basics first; continued proportion extends it to the mean proportional.
3. 3Properties of Proportion: Componendo and DividendoThese are shortcuts that work on a given proportion, so learn them after you can handle proportion itself.
4. 4Variation and Problems on RatioVariation and word problems use ratio and proportion together, so they come once the tools are ready.
5. 5Laws of IndicesLogarithm laws mirror index laws, so master powers before moving to logs.
6. 6Logarithms: Definition and LawsA logarithm is the inverse of a power; the definition and laws follow directly from the index laws.
7. 7Common and Natural Logarithms and Change of BaseThis last topic applies the log laws to base 10 and base e, and to switching bases, so it needs all earlier logs.

How to prepare Ratio and Proportion, Indices and Logarithms

1. Write a one-page rule sheet while you study: ratio types, proportion conditions, componendo and dividendo, index laws, log laws and change of base.
2. For each rule, solve three to five short questions straight away, so the rule becomes automatic.
3. For ratio and variation word problems, assign a common multiplier k (for example, quantities as 3k and 5k) and form one equation.
4. For indices and logs, simplify in small steps, one law per line, and keep the base the same before combining.
5. Practise MCQs under time: aim for about a minute each, and use option elimination or substituting simple values when a question looks long.
6. Skip a question that needs more than two minutes on the first pass, because wrong answers cost 0.25 marks each. Return to it if time remains.
7. Revise your rule sheet every few days, and do a mixed set of questions from all seven topics before the exam.

Common mistakes in Ratio and Proportion, Indices and Logarithms

• Writing log(m + n) as log m + log n.

  Fix: Remember the law is for products only: log(mn) = log m + log n. There is no simple rule for log(m + n).

• Combining indices when the bases are different, such as 2³ × 3² = 6⁵.

  Fix: Add powers only when the base is the same. Rewrite numbers to a common base first, for example 4 as 2².

• Using componendo and dividendo in the wrong direction or without checking the proportion is set up as a/b = c/d.

  Fix: Write a/b = c/d first, then apply the rule term by term. Check that the denominator you divide by is not zero.

• Taking the logarithm of zero or a negative number, or using base 1.

  Fix: Recall that logₐ x needs a > 0, a ≠ 1 and x > 0. Check these conditions in equations before accepting an answer.

• Adding the ratio terms to find the share without using the total, or mixing units in a ratio.

  Fix: Convert to the same unit, write quantities as ak and bk, and use the given total or difference to find k.

• Confusing direct and inverse variation.

  Fix: Ask: if one doubles, does the other double (direct, y = kx) or halve (inverse, xy = k)? Then set up the equation.

Last-day revision: Ratio and Proportion, Indices and Logarithms

• ✓Ratio a : b = a ÷ b, with b ≠ 0; it compares quantities of the same kind in the same unit.
• ✓Multiplying or dividing both terms of a ratio by the same non-zero number does not change it.
• ✓Proportion: a : b = c : d means ad = bc.
• ✓Continued proportion a : b = b : c gives b² = ac, so b = √(ac) is the mean proportional.
• ✓Componendo: if a/b = c/d then (a + b)/b = (c + d)/d.
• ✓Dividendo: if a/b = c/d then (a − b)/b = (c − d)/d.
• ✓Componendo-dividendo: if a/b = c/d then (a + b)/(a − b) = (c + d)/(c − d), where the denominators are non-zero.
• ✓Indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 for a ≠ 0.
• ✓Negative index: a⁻ⁿ = 1/aⁿ; fractional index: a^(1/n) is the nth root of a.
• ✓Logarithm: logₐ x = y means aʸ = x, for a > 0, a ≠ 1 and x > 0.
• ✓Log laws: log(mn) = log m + log n; log(m/n) = log m − log n; log(mⁿ) = n log m; logₐ 1 = 0; logₐ a = 1.
• ✓Change of base: logₐ b = log b ÷ log a (any common base) = 1 ÷ logᵦ a.

Ratio and Proportion, Indices and Logarithms practice questions

• If log₂x + log₄x = 6, then the value of x is:
• The ratio compounded of 2:3, 9:4 and 5:6 is:
• If A : B = 3 : 5 and B : C = 2 : 7, then the ratio A : B : C is:
• The value of log to the base 4 of 32 is:
• If 2^x = 3^y = 6^(−z), then the value of 1/x + 1/y + 1/z is:
• The mean proportional between 4 and 36 is:
• A sum of money is divided among Priya, Quincy, and Ravi in the ratio 5:7:8. If Ravi receives ₹4,800 more than Priya, what is the total sum d…
• A variable Y is inversely proportional to the square of variable X. When X = 2, Y = 18. What will be the value of Y when X = 3?