What is 10/31 plus 9/13? (Answer: 409/403)

Exact Answer

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Problem: What is 10/31 plus 9/13?
Formula / Expression: 10/31 + 9/13
Final Answer: 1 6/403 (409/403)
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Step-by-Step Solution:

Step 1: Find the Least Common Denominator (LCD)
The denominators 31 and 13 are coprime (they share no common factors other than 1). Therefore, their Least Common Denominator (LCD) is simply their direct product: 31 × 13 = 403.
Working: LCD(31, 13) = 403

Step 2: Convert to Equivalent Fractions with Common Denominator 403
Multiply the numerator and denominator of 10/31 by 13, and multiply the numerator and denominator of 9/13 by 31 so both fractions share the common base 403.
Working: 10 × 13/31 × 13 = 130/403 and 9 × 31/13 × 31 = 279/403

Step 3: Add the Numerators
Now add the numerators (130 + 279 = 409) and keep the common denominator 403 unchanged.
Working: 130 + 279/403 = 409/403

Step 4: Check if the Fraction Can Be Simplified
The Greatest Common Divisor (GCD) of 409 and 403 is 1, confirming that 409/403 cannot be reduced further—it is already in its simplest irreducible form.
Working: 409/403

Step 5: Convert Improper Fraction to a Mixed Number
Since the numerator (409) is greater than the denominator (403), divide 409 by 403: it goes in 1 whole time(s) with a remainder of 6, giving the mixed number 1 6/403.
Working: 409/403 = 1 6/403

Step 6: Decimal & Percentage Representation
Dividing the numerator 409 by denominator 403 produces a repeating decimal (approximately 1.014888, or 101.4888%).
Working: 409 ÷ 403 = 1.014888 (101.4888%)

Key Values:
- Simplified Fraction: 409/403
- Mixed Number Form: 1 6/403
- Decimal Value: 1.014888
- Percentage: 101.4888%

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Solved with FractionsCalc:
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10/31 + 9/13 = 409/403

Simplified Fraction

409/403

Mixed Number

1 6/403

Decimal Value

1.014888

Percentage

101.4888%

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Step-by-Step Solution: 10/31 + 9/13

Step 1

Find the Least Common Denominator (LCD)

The denominators 31 and 13 are coprime (they share no common factors other than 1). Therefore, their Least Common Denominator (LCD) is simply their direct product: 31 × 13 = 403.

LCD(31, 13) = 403

Step 2

Convert to Equivalent Fractions with Common Denominator 403

Multiply the numerator and denominator of 10/31 by 13, and multiply the numerator and denominator of 9/13 by 31 so both fractions share the common base 403.

10 × 13/31 × 13 = 130/403 and 9 × 31/13 × 31 = 279/403

Step 3

Add the Numerators

Now add the numerators (130 + 279 = 409) and keep the common denominator 403 unchanged.

130 + 279/403 = 409/403

Step 4

Check if the Fraction Can Be Simplified

The Greatest Common Divisor (GCD) of 409 and 403 is 1, confirming that 409/403 cannot be reduced further—it is already in its simplest irreducible form.

409/403

Step 5

Convert Improper Fraction to a Mixed Number

Since the numerator (409) is greater than the denominator (403), divide 409 by 403: it goes in 1 whole time(s) with a remainder of 6, giving the mixed number 1 6/403.

409/403 = 16/403

Step 6

Decimal & Percentage Representation

Dividing the numerator 409 by denominator 403 produces a repeating decimal (approximately 1.014888, or 101.4888%).

409 ÷ 403 = 1.014888 (101.4888%)

Visual Representation

The diagram below illustrates 10/31 plus 9/13 visually:

Summary & Alternative Representations

Property / Format	Value
Lowest Terms Fraction	`409/403`
Mixed Number Form	`1 6/403`
Decimal Form	`1.014888`
Percentage	`101.4888%`

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Frequently Asked Questions

What is 10/31 plus 9/13 as a fraction?

10/31 + 9/13 in simplest fraction form is 409/403 (or 1 6/403 as a mixed number).

What is 10/31 plus 9/13 as a decimal?

10/31 + 9/13 converted to a decimal is 1.014888.

What is 10/31 plus 9/13 as a percentage?

10/31 + 9/13 expressed as a percentage is 101.4888%.

How do you write 10/31 plus 9/13 as a mixed number?

Because the numerator is greater than the denominator, the improper fraction 409/403 can be written as the mixed number 1 6/403.

Does 10/31 plus 9/13 result in a terminating or repeating decimal?

It results in a repeating decimal (approximately 1.014888...) because the denominator contains prime factors other than 2 and 5.