Multiplying the common ordinary fractions: - ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} = ? The multiplication process explained. The result written: As a negative proper fraction (the numerator < the denominator). As a decimal number. As a percentage
- ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} = ?
Simplify the operation
Rewrite the equivalent simplified operation:
Combine the signs of the fractions into a single one, placed in front of the expression. If the sign is + then it is usually not written.
The sign of a multiplication operation:
+ 1 × + 1 = + 1
+ 1 × - 1 = - 1
- 1 × - 1 = + 1
- ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} =
- ^{39}/_{30} × ^{36}/_{53} × ^{31}/_{141}
Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:
To fully reduce a fraction, to the lowest terms equivalent: divide the numerator and denominator by their greatest common factor, GCF.
* Why do we try to reduce (simplify) the fractions?
By reducing the values of the numerators and the denominators of the fractions the calculations are easier to make.
A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.
* In order to easily reduce a fraction, factor its numerator and denominator. This way all the common prime factors are easily identified and crossed out, without calculating the GCF.
^{39}/_{30} =
^{(3 × 13)}/_{(2 × 3 × 5)} =
^{((3 × 13) ÷ 3)}/_{((2 × 3 × 5) ÷ 3)} =
^{(3 ÷ 3 × 13)}/_{(2 × 3 ÷ 3 × 5)} =
^{(1 × 13)}/_{(2 × 1 × 5)} =
^{13}/_{10}
^{36}/_{53} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors.
Their prime factorization:
36 = 2^{2} × 3^{2}
53 is a prime number (it cannot be factored into other prime factors)
^{31}/_{141} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors.
Their prime factorization:
31 is a prime number (it cannot be factored into other prime factors)
141 = 3 × 47
Rewrite the equivalent simplified operation:
- ^{39}/_{30} × ^{36}/_{53} × ^{31}/_{141} =
- ^{13}/_{10} × ^{36}/_{53} × ^{31}/_{141}
Perform the operation of calculating the fractions
Multiply the fractions:
1) Multiply the numerators, that is, all the numbers above the fractions bars, separately.
2) Multiply the denominators, that is, all the numbers below the fractions bars, separately.
* Factor all the numerators and all the denominators in order to easily reduce (simplify) the end fraction.
- ^{13}/_{10} × ^{36}/_{53} × ^{31}/_{141} =
- ^{(13 × 36 × 31)} / _{(10 × 53 × 141)} =
- ^{(13 × 22 × 32 × 31)} / _{(2 × 5 × 53 × 3 × 47)} =
- ^{(22 × 32 × 13 × 31)} / _{(2 × 3 × 5 × 47 × 53)}
Fully reduce (simplify) the end fraction to its lowest terms equivalent:
Calculate the greatest common factor, GCF,
of the numerator and denominator of the fraction:
A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.
To fully reduce a fraction, to the lowest terms equivalent: divide the numerator and denominator by their greatest common factor, GCF.
* To calculate the GCF, we need to factor the numerator and the denominator of the fraction into prime factors.
Then multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).
GCF (2^{2} × 3^{2} × 13 × 31; 2 × 3 × 5 × 47 × 53) = 2 × 3
Divide the numerator and the denominator by their GCF:
- ^{(22 × 32 × 13 × 31)} / _{(2 × 3 × 5 × 47 × 53)} =
- ^{((22 × 32 × 13 × 31) ÷ (2 × 3))} / _{((2 × 3 × 5 × 47 × 53) ÷ (2 × 3))} =
- ^{(22 ÷ 2 × 32 ÷ 3 × 13 × 31)}/_{(2 ÷ 2 × 3 ÷ 3 × 5 × 47 × 53)} =
- ^{(2(2 - 1) × 3(2 - 1) × 13 × 31)}/_{(1 × 1 × 5 × 47 × 53)} =
- ^{(21 × 31 × 13 × 31)}/_{(1 × 1 × 5 × 47 × 53)} =
- ^{(2 × 3 × 13 × 31)}/_{(1 × 1 × 5 × 47 × 53)} =
- ^{(2 × 3 × 13 × 31)}/_{(5 × 47 × 53)} =
- ^{2,418}/_{12,455}
Rewrite the fraction
As a decimal number:
Simply divide the numerator by the denominator, without a remainder, as shown below:
- ^{2,418}/_{12,455} =
- 2,418 ÷ 12,455 ≈
- 0.19413890004 ≈
- 0.19
As a percentage:
A percentage value p% is equal to the fraction: ^{p}/_{100}, for any decimal number p. So, we need to change the form of the number calculated above, to show a denominator of 100.
To do that, multiply the number by the fraction ^{100}/_{100}.
The value of the fraction ^{100}/_{100} = 1, so by multiplying the number by this fraction the result is not changing, only the form.
- 0.19413890004 =
- 0.19413890004 × ^{100}/_{100} =
^{( - 0.19413890004 × 100)}/_{100} =
^{ - 19.413890004014}/_{100} ≈
- 19.413890004014% ≈
- 19.41%
The final answer:
written in three ways
As a negative proper fraction:
(the numerator < the denominator)
- ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} = - ^{2,418}/_{12,455}
As a decimal number:
- ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} ≈ - 0.19
As a percentage:
- ^{39}/_{30} × - ^{36}/_{53} × - ^{31}/_{141} ≈ - 19.41%
How are the numbers being written on our website: comma ',' is used as a thousands separator; point '.' used as a decimal separator; numbers rounded off to max. 12 decimals (if the case). The set of the used symbols on our website: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.
Other similar operations
Multiply common ordinary fractions, online calculator:
The latest common ordinary fractions that have been multiplied
Multiplying fractions. How to multiply ordinary math fractions? Steps. Example.
How to multiply two fractions?
When we multiply ordinary fractions, the end fraction will have:
- as a numerator, the result of multiplying all the numerators of the fractions,
- as a denominator, the result of multiplying all the denominators of the fractions.
- ^{a}/_{b} × ^{c}/_{d} = ^{(a × c)} / _{(b × d)}
- a, b, c, d are integer numbers;
- if the pairs (a × c) and (b × d) are not coprime (they have common prime factors) the end fraction should be reduced (simplified) to lower terms.
How to multiply ordinary fractions? Steps.
- Start by reducing fractions to lower terms (simplifying).
- Internal link > Reduce common ordinary fractions to the lowest terms, online, with explanations
- Factor the numerators and the denominators of the reduced fractions: break them down to their prime factors.
- External link > Check whether numbers are prime or not. Calculate the prime factors of the composite numbers, online calculator.
- Above the fraction bar we write the product of all the prime factors of the fractions' numerators, without doing any calculations.
- Below the fraction bar we write the product of all the prime factors of the fractions' denominators, without doing any calculations.
- Cross out all the common prime factors that appear both above and below the fraction bar.
- Multiply the remaining prime factors above the fraction bar - this will be the numerator of the resulted fraction.
- Multiply the remaining prime factors below the fraction bar - this will be the denominator of the resulted fraction.
- There is no need to reduce (simplify) the resulting fraction, since we have already crossed out all the common prime factors.
- If the resulted fraction is an improper one (without considering the sign, the numerator is larger than the denominator), it could be written as a mixed number, consisting of an integer and a proper fraction of the same sign.
- Internal link > Reduce (simplify) and write improper fractions as mixed numbers, online calculator
- Internal link > Multiply common ordinary fractions, online, with explanations
More on ordinary (common) fractions / theory: