Ladder Fraction: Formula, Examples and Solving Methods
Ladder Fraction is a nested fraction in which one fraction appears inside the denominator of another. These expressions are simplified from the innermost bracket outward. This page explains the ladder fraction formula, the bottom-up method, useful shortcuts, solved ladder fraction examples, and common rules for aptitude questions.
What Is a Ladder Fraction?
For a ladder fraction with terms a₁, a₂, ..., aₙ, define the last value as xₙ = aₙ. Then calculate xᵢ = aᵢ + 1/xᵢ₊₁ for i = n − 1, n − 2, ..., 1. The final value x₁ is the simplified fraction. For example, 1 + 1/(2 + 1/3) = 1 + 1/(7/3) = 1 + 3/7 = 10/7.
Ladder Fraction Video Lessons
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Ladder Fractions in Simplification
Learn how to simplify ladder fractions by evaluating nested fractional expressions step by step and applying the correct order of operations.
Ladder Fraction Formula & Tricks
Important Formulas
The nested expression is evaluated from the innermost denominator toward the outside.
Start with the last term and calculate each preceding value one at a time.
Since b + 1/c = (bc + 1)/c, its reciprocal is c/(bc + 1).
Quick Tricks
Do not begin with the outermost fraction. First combine the deepest denominator, take its reciprocal, and continue outward.
After every step, reduce the fraction before continuing. This keeps the next reciprocal calculation short and reduces sign errors.
The reciprocal of p/q is q/p, provided p and q are non-zero. The reciprocal applies to the complete bracketed denominator.
Ladder Fraction Concepts
Bottom-Up Simplification Method
For each level, perform the addition or subtraction inside the denominator before taking the reciprocal. In 3 + 1/(4 + 1/2), first calculate 4 + 1/2 = 9/2. Therefore, the expression becomes 3 + 2/9 = 29/9.
Converting a Nested Denominator
This conversion removes one level of nesting. For non-zero c, b + 1/c = (bc + 1)/c. Thus, 1/(b + 1/c) = c/(bc + 1).
Ladder Fractions with Subtraction
The same bottom-up method applies to expressions such as a − 1/(b − 1/c). First calculate b − 1/c, then take its reciprocal, and finally apply the outside subtraction. For example, 2 − 1/(3 − 1/2) = 2 − 1/(5/2) = 2 − 2/5 = 8/5.
Repeated Unit Ladder Fractions
For Fₙ = 1 + 1/(1 + 1/(... + 1)) with n terms, the value is Fₙ = Fib(n + 2)/Fib(n + 1), where Fib(1) = 1 and Fib(2) = 1. For three terms, the value is 1 + 1/(1 + 1/1) = 1 + 1/2 = 3/2.
Ladder Fraction Data Table
Use the structured data below while following the explanation and examples.
Bottom-Up Evaluation of a Ladder Fraction
Intermediate values for simplifying 2 + 1/(3 + 1/4) from the innermost level outward.
| Step | Expression or value | Result |
|---|---|---|
| 1 | 1/4 | 1/4 |
| 2 | 3 + 1/4 | 13/4 |
| 3 | 1 ÷ (13/4) | 4/13 |
| 4 | 2 + 4/13 | 30/13 |
Note: The final simplified value is 30/13.
Practice Ladder Fraction Questions
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Ladder Fraction Revision Points
Use these rules to simplify nested fractions accurately.
- A ladder fraction is evaluated from the innermost denominator outward.
- For x₁ = a₁ + 1/(a₂ + ... + 1/aₙ), set xₙ = aₙ and use xᵢ = aᵢ + 1/xᵢ₊₁.
- The reciprocal of p/q is q/p, not p/q.
- Always apply the reciprocal to the complete bracketed denominator.
- For subtraction, preserve every sign while working from the inside outward.
- For n repeated 1s, the value is Fib(n + 2)/Fib(n + 1).
Ladder Fraction FAQs
What is the ladder fraction formula?
For x₁ = a₁ + 1/(a₂ + 1/(... + 1/aₙ)), set xₙ = aₙ and calculate xᵢ = aᵢ + 1/xᵢ₊₁ from right to left.
How do you solve 1 + 1/(2 + 1/3)?
First, 2 + 1/3 = 7/3. Therefore, 1 + 1/(7/3) = 1 + 3/7 = 10/7.
What is the value of 2 + 1/(3 + 1/4)?
3 + 1/4 = 13/4, so 1/(13/4) = 4/13. Hence, the value is 2 + 4/13 = 30/13.
Is 1/(a + 1/b) equal to 1/a + 1/b?
No. The reciprocal applies to the complete denominator: 1/(a + 1/b) = b/(ab + 1), for non-zero b.
How is subtraction handled in a ladder fraction?
Keep the signs unchanged and simplify from the innermost level. For example, 2 − 1/(3 − 1/2) = 2 − 1/(5/2) = 8/5.
What is the repeated-unit ladder fraction with three 1s?
It is 1 + 1/(1 + 1/1) = 1 + 1/2 = 3/2.