Answer to Question #342924 in Algebra for Lebu

Question

Use a standard algorithm to calculate: $4 \frac{2}{3}+\frac{2}{3}-1\frac{3}{4}$ . Explain how you worked on the second step to assist the understanding of the standard algorithm used.

4 and 1are whole numbers and the rest are fractions.

Answer 1

Mixed fraction is a fraction that is written as a combination of a natural number and a proper fraction. It is a simplified form of an improper fraction. In this example we have two mixed fractions: $4\frac 2 3$ and $1\frac 3 4$

To calculate this sample we need to convert a mixed number to an improper fraction. To do that we need to multiply the whole number with the denominator and then add this product with the numerator. This forms the new numerator of the improper fraction while the denominator remains the same.

So we have: $4\frac 2 3=\frac {3*4+2} 3=\frac {14} 3$ and $1\frac 3 4=\frac {4*1+3} 4=\frac 7 4$ .

Thus we have: $\frac {14} 3+\frac 2 3-\frac 7 4$

Standard algorithm for adding fractions: Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. (In general, 𝘢/𝘣 + 𝘤/𝘥 = (𝘢𝘥 + 𝘣𝘤)/𝘣𝘥).

So, algorithm to add two fractions

1. Find a common denominator by finding the LCM (Least Common Multiple) of the two denominators.
2. Change the fractions to have the same denominator and add both terms.

Fractions with the same denominators are called like fractions . To add fractions with like denominators, add the numerators , and write the sum over the denominator.

Hence we have: $\frac {14} 3+\frac 2 3-\frac 7 4=\frac {14+2} 3-\frac 7 4=\frac {16} 3-\frac 7 4=\frac {16*4-7*3}{3*4}=\frac {43}{12}$

And now we can convert improper fraction to mixed number. To do that we should:

1. Divide the numerator by the denominator
2. Write down the whole number result
3. Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number.

So,

1. Divide 43 by 12: 43 ÷ 12 = 3 with remainder of 7
2. The whole number result is 3
3. The remainder is 7. With 7 as the numerator and 12 as the denominator, the fraction part of the mixed number is 7/12.

So, $4\frac 2 3 +\frac 2 3-1\frac 3 4=\frac {43} {12}=3\frac 7 {12}$