Question

Without using a calculator find the value of $$ 3 \frac{13}{17}+\frac{4-\frac{1}{3}}{3 / 7} $$

Short answer

Question: Simplify the given expression without using a calculator: $$3 \frac{13}{17}+\frac{4-\frac{1}{3}}{3 / 7}$$ Answer: $$\frac{1885}{153}$$

Step-by-step solution

Convert the mixed number into an improper fraction

Convert the mixed number $$3 \frac{13}{17}$$ into an improper fraction by multiplying the whole number by the denominator, then adding the numerator and placing that value over the denominator. So we get: $$ 3 \frac{13}{17} = \frac{3 \cdot 17 + 13}{17} = \frac{51 + 13}{17} = \frac{64}{17} $$

Simplify the expression within the parenthesis of the second part

Next, we will simplify the expression inside the parenthesis $$4-\frac{1}{3}$$ by finding a common denominator and then subtracting the fractions: $$ 4-\frac{1}{3} = \frac{12}{3}-\frac{1}{3} = \frac{11}{3} $$

Replace the parenthesis with the simplified expression

Now, we need to substitute the simplified expression found in Step 2 into the original expression. So we get: $$ 3 \frac{13}{17}+\frac{4-\frac{1}{3}}{3 / 7} = \frac{64}{17} + \frac{\frac{11}{3}}{3 / 7} $$

Simplify the complex fraction

In this step, we will simplify the complex fraction in the second part. To do this, we will multiply the numerator by the reciprocal of the denominator: $$ \frac{\frac{11}{3}}{3 / 7} = \frac{11}{3} \cdot \frac{7}{3} = \frac{11 \cdot 7}{3 \cdot 3} = \frac{77}{9} $$

Add the simplified fractions

Finally, we will add the two simplified fractions found in Steps 1 and 4. To do this, we need to first find the common denominator, which in this case is $$17 \cdot 9$$: $$ \frac{64}{17} + \frac{77}{9} = \frac{64 \cdot 9}{17 \cdot 9} + \frac{77 \cdot 17}{9 \cdot 17} = \frac{576 + 1309}{153} = \frac{1885}{153} $$ So the simplified value of the given expression is $$\frac{1885}{153}$$.

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the original exercise, you see the mixed number \(3 \frac{13}{17}\). This means you have the whole number 3 and the fraction \(\frac{13}{17}\).

To make calculations simpler, it's often necessary to convert mixed numbers into improper fractions. Improper fractions have a numerator that is larger than or equal to the denominator. This makes it easier to perform operations like addition or subtraction. In our example, \(3 \frac{13}{17}\) converts into an improper fraction by multiplying 3 by 17 (the denominator) and then adding 13 (the numerator):

\[ 3 \frac{13}{17} = \frac{(3 \times 17) + 13}{17} = \frac{64}{17} \]

Once converted, improper fractions can be easily used in more complex mathematical expressions.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator, such as \(\frac{64}{17}\) in the provided exercise. These types of fractions are helpful when performing arithmetic operations like addition, subtraction, multiplication, and division compared to mixed numbers.

One of the reasons improper fractions are favored in calculations is they keep everything as a single fraction, thus avoiding confusion and mistakes that can arise from juggling multiple parts (whole numbers and fractions).

For example, if during a calculation you need to add or multiply fractions, improper fractions allow you to easily find common denominators or simply multiply across the numerators and denominators without worrying about the whole number part. This consistency and simplicity is what makes them extremely useful in fractions and simplifies problem-solving.

Complex Fractions

Complex fractions involve fractions within fractions, often known as a "fraction of fractions." In the original exercise, a complex fraction is present in the form \(\frac{\frac{11}{3}}{\frac{3}{7}}\).

To simplify complex fractions, you need to multiply the numerator by the reciprocal of the denominator. The reciprocal is simply flipping the fraction. So, \(\frac{3}{7}\) becomes \(\frac{7}{3}\).

In our case, simplify the complex fraction by:

\[ \frac{\frac{11}{3}}{\frac{3}{7}} = \frac{11}{3} \times \frac{7}{3} = \frac{11 \times 7}{3 \times 3} = \frac{77}{9} \]

This simplification process allows complex fractions to be converted into a straightforward, single fraction, making further calculations easier.

Simplifying Expressions

Simplifying expressions involves making an expression as straightforward and manageable as possible. In the original math problem, you have the expression \(3 \frac{13}{17} + \frac{\frac{11}{3}}{\frac{3}{7}}\) which looks complex initially.

The process involves several steps:

• Convert mixed numbers to improper fractions.
• Simplify any complicated parts, such as fractions within fractions.
• Find a common denominator if you need to add or subtract fractions.

The aim is to get a single simplified fraction. In our case, the solution involves first turning \(3 \frac{13}{17}\) into \(\frac{64}{17}\), and simplifying \(\frac{\frac{11}{3}}{\frac{3}{7}}\) to \(\frac{77}{9}\).

Finally, the two results are added. To do this, we need a common denominator, which ensures the fractions combine seamlessly into one single expression:

\[ \frac{64}{17} + \frac{77}{9} = \frac{64 \times 9}{17 \times 9} + \frac{77 \times 17}{9 \times 17} = \frac{576 + 1309}{153} = \frac{1885}{153} \]

These steps help you turn complex expressions into something more manageable and readable.