Fractions
Calculation has two complex categories related to fractions.
These categories involve a series of terms of the form x / y, which are either:
added together, or
multiplied together
In some cases, the end of the question includes a “× z”, which increases the level of accuracy required for the preceding calculation.
Example 1
(82/25) + (98/50) + (89/57) + (34/80) + (94/91)
(easy level)
This problem involves adding several fractions. Because the final answer is less than 10, we only need two significant digits.
We could calculate each fraction to two significant digits and accept some risk of rounding error, or calculate each to three significant digits to greatly reduce that risk.
For the first, second, and fourth terms, the denominators are “nice” numbers that allow us to convert the fractions into values out of 100.
(82/25) × (4/4) = 328/100 → 3.28
(98/50) × (2/2) = 196/100 → 1.96
(34/80) × (1.25/1.25) = 42.5/100 → 0.425
The sum of these three terms is:
3.28 + 1.96 + 0.425 = 5.665
The remaining terms can be handled with straightforward division:
89/57 → 1.56
94/91 → 1.03
These sum to approximately 2.59.
Adding this to our prior total gives:
5.665 + 2.59 = 8.255
This is already correct to more than the required number of significant digits.
Example 2
((41/13) × (86/77) × (70/78) × (92/23) × (90/27)) × 38
(easy level)
This is a difficult question for the easy level. Because of the final × 38, and because the product of the five fractions exceeds 10, we need four significant digits for the product of the fractions.
The first step is to simplify.
92/23 = 4
90/27 = 10/3
So the last two terms combine to 40/3.
Next, notice that the second and third fractions are close to cancelling:
86 × 70 = 6020
77 × 78 = 6006
Their product is very close to 1. However, since we need four significant digits, we cannot simply cancel them outright.
Instead, reduce by common factors:
(86 × 70) / (77 × 78)
→ (43 × 10) / (11 × 39)
→ 430/429
Now the expression becomes:
(40/3) × (41/13) × 38 × (430/429)
This simplifies to:
(40 × 41 × 38) / 39 × (slight adjustment)
We know:
(40 × 38) / 39 is slightly less than 39
In fact, it is 38 and 38/39, which is 39 − 1/39
Now:
39 × 41 is one less than 40² → 1599
We need to:
adjust down by 1 because 40 × 38 is one less than 39²
adjust up by about 4 due to the 430/429 term
This leads to a final answer of approximately:
1602
Example 3
(946/169) + (858/290) + (521/106) + (623/542) + (481/160)
(advanced level)
In Calculation League, fraction-addition questions without a final multiplication only require two significant digits.
When only two significant digits are needed, rounding and estimation — while loosely tracking the direction of rounding — is usually the most efficient approach.
Approximate each term:
946/169 → 5.6
858/290 → 2.95
521/106 → 5.0
623/542 → 1.15
481/160 → 3.0
These estimates sum to:
17.7
If speed is prioritized, we could round each to the nearest integer:
6 + 3 + 5 + 1 + 3 = 18
This is sufficient, especially since several terms are already close to integers, minimizing rounding risk.
Example 4
((249/364) × (323/258) × (418/560)) × 5
(advanced level)
Despite being labeled “advanced,” this question is easier than Example 2.
Rearrange and scan for simplifications:
(249 × 323 × 418 × 5) / (364 × 258 × 560)
Notice that 249 and 258 are close. Cancel them and compensate by multiplying another term by 258/249, which is 1 + 9/249.
Since 249/9 = 83/3, this is equivalent to adding 13 to 364:
364 → 377
Now the expression becomes:
(323 × 418 × 5) / (377 × 560)
Reduce:
5/560 = 1/112
So we have:
(323 × 418) / (377 × 112)
Estimate:
418/377 → 1.11
323/112 → 2.88
Multiplying:
1.11 × 2.88 ≈ 3.20
This is correct to three significant digits, one more than required.
Example 5
((4126/5383) + (8850/7162) + (6877/1696)) × 29
(expert level)
At expert level, the size of the numbers and the final multiplication force us to perform more accurate division.
Nearest 0.1
4126/5383 → 0.8
8850/7162 → 1.2
6877/1696 → 4.1
Sum = 6.1
6.1 × 29 = 176.9
Nearest 0.01
4126/5383 → 0.77
8850/7162 → 1.24
6877/1696 → 4.05
Sum = 6.06
6.06 × 29 = 175.74
Nearest 0.001
4126/5383 → 0.766
8850/7162 → 1.236
6877/1696 → 4.055
Sum = 6.057
6.057 × 29 = 175.653
Correct answer: 175.653
Notes on Error Control
When adding fractions:
The number of fractions does not increase average error.
Rounding to the nearest 0.1 produces an average error of 0.025, regardless of how many fractions are added.
Multiplying by 29 scales that error to about 0.725, which is large enough to matter but small enough to refine.
Two adjustment strategies:
Track rounding direction — here we rounded up twice and down once, suggesting a slight upward bias.
Use halfway points when a value is clearly closer to the midpoint.
Example 6
((334/440) × (403/540) × (159/877)) × 9
(expert level)
This has the same structure as the advanced-level multiplication question.
Rearrange:
(334 × 403 × 159 × 9) / (440 × 540 × 877)
Reduce:
9/540 = 1/60
Now:
(334 × 403 × 159) / (60 × 440 × 877)
Notice:
60 × 440 = 26,400
Estimating 334 × 159 as 330 × 160 = 52,800, which is exactly double 26,400
So:
(334 × 159) / (440 × 60) ≈ 2
Remaining term: 403/877
Doubling the numerator:
806/877 ≈ 0.92
This is already correct to two decimal places.
For more precision:
Adjust the “2” upward slightly (about 1.1–1.2%)
This adds roughly 4 to 806 → 810
810/877 ≈ 0.924
This is correct to three significant digits.