Question

Find the sum. $$1 \frac{7}{9}+4 \frac{3}{7}$$

Short answer

The sum of \(1 \frac{7}{9}+4 \frac{3}{7}\) is \(6\frac{13}{63}\)

Step-by-step solution

Split and Add Whole Numbers

First, separate the whole number part and the fraction part in both numbers. Add the whole numbers together. So, \(1+4 = 5\)

Add Fractions

Next, add the fractions. Fractions need to have a common denominator in order to be added together. So, \(\frac{7}{9}+\frac{3}{7}\) needs to be converted into fractions with the same denominator. A common multiple of 9 and 7 is 63. Therefore, the denominators should be converted to 63. Hence, we get \(\frac{49}{63}+\frac{27}{63}=\frac{76}{63}\)

Simplify the Fraction

In Step 2, we got the result as \(\frac{76}{63}\). But this is an improper fraction (numerator > denominator). Therefore, we have to convert it to a proper form. \(\frac{76}{63}\) can be written as \(1\frac{13}{63}\), as 76 divided by 63 equals 1 remainder 13.

Combine the Whole Number and Fraction

Finally, add the whole number sum from Step 1 and the fractional part from Step 3. So, \(5 + 1\frac{13}{63} = 6\frac{13}{63}\)

Key concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is equal to or larger than the denominator (the bottom number). For example, in the exercise, after adding the fractions, we get \( \frac{76}{63} \), which is an improper fraction because 76 (the numerator) is greater than 63 (the denominator).

Converting an improper fraction into a mixed number helps in understanding the value of the fraction in terms of whole parts and a fractional remainder. To do this, we divide the numerator by the denominator. In the given example, dividing 76 by 63 gives us 1 whole part and a remainder of 13, so \( \frac{76}{63} \) becomes \( 1\frac{13}{63} \). This conversion is crucial for understanding and performing further arithmetic operations, such as combining it with a whole number as seen in the final step of the exercise.

Common Denominator

A common denominator is required when adding or subtracting fractions that have different denominators, as in the exercise \(\frac{7}{9} + \frac{3}{7}\). The common denominator is a common multiple of the original denominators (in this case, 9 and 7).

To find a common denominator, one can list out the multiples of each denominator and choose the smallest multiple that appears on both lists. Alternatively, multiplying the denominators together guarantees a common multiple but not necessarily the smallest. In the exercise, 63 is identified as a common multiple of 9 and 7. The fractions are then converted to this common denominator before they are added, resulting in \(\frac{49}{63} + \frac{27}{63}\).

Simplify Fractions

To simplify a fraction, the goal is to make the fraction as straightforward as possible, usually by ensuring that the numerator and the denominator have no common divisors other than 1. Simplifying does not change the value of the fraction but makes it easier to understand or further manipulate.

In the context of the exercise, once we have added the fractions \(\frac{49}{63} + \frac{27}{63} = \frac{76}{63}\), we have an improper fraction which doesn't need simplification in terms of reducing it to lower terms, but instead, it's expressed in mixed number form to provide a clearer understanding of its value. This step is crucial in obtaining a final answer that's easy to read and understand.