Problem 80 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$3 \frac{5}{10} \times \frac{4}{100}$$

Step-by-Step Solution

Answer

The product is \(\frac{7}{50}\).

Step 1: Convert Mixed Number to Improper Fraction

To simplify the multiplication of mixed numbers and fractions, convert the mixed number into an improper fraction. The mixed number is given as \(3 \frac{5}{10}\). First convert \(\frac{5}{10}\) to its simplest form: \(\frac{1}{2}\). So the mixed number is \(3 \frac{1}{2}\). Now convert this to an improper fraction: \(3 \frac{1}{2} = \frac{7}{2}\).

Step 2: Set Up the Multiplication Problem

Now that you have the improper fraction \(\frac{7}{2}\), set up the multiplication with the other fraction \(\frac{4}{100}\). The new multiplication problem is \(\frac{7}{2} \times \frac{4}{100}\).

Step 3: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together. Numerators: \(7 \times 4 = 28\). Denominators: \(2 \times 100 = 200\). This gives \(\frac{28}{200}\).

Step 4: Simplify the Fraction

Reduce \(\frac{28}{200}\) to its simplest form. Find the greatest common divisor of 28 and 200, which is 4. Divide the numerator and the denominator by 4: \(\frac{28 \div 4}{200 \div 4} = \frac{7}{50}\).

Step 5: Final Answer

The product of the given fractions is \(\frac{7}{50}\). This is our final simplified answer.

Key Concepts

Improper Fractions

Improper fractions are an essential concept when dealing with fractions and mixed numbers. They are called "improper" because the numerator, which is the top number, is larger than the denominator, the bottom number. This might seem a bit odd at first, but it’s quite useful! When we multiply fractions, especially when dealing with mixed numbers, converting to improper fractions is handy. To do this, for a mixed number like 3 1/2, you multiply the whole number part 3 by the denominator 2, giving 6, and then add the numerator 1 to get 7. So, 3 1/2 as an improper fraction is 7/2. This form allows easier multiplication and division, as you merely handle one fraction form rather than switching between mixed and improper formats.

Mixed Numbers

Mixed numbers have both a whole number and a fractional part, like 3 1/2. These are often used in daily life because they seem more intuitive than improper fractions. However, they aren't as straightforward when performing arithmetic operations like multiplication. That's why converting mixed numbers into improper fractions can make calculations simpler. To convert a mixed number into an improper fraction, use the process mentioned before: multiply the whole number by the fraction's denominator then add the numerator. This step helps maintain consistency when multiplying or dividing, ensuring you're working with one form.

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible. It means reducing the fraction to its smallest basic form where the numerator and denominator share no common factors other than 1. For example, given a fraction like 28/200, recognize that both numbers can be divided by the greatest common divisor, which is 4 here. Divide both by 4 to get the simplified fraction 7/50. Simplification makes fractions easier to understand and work with. It often provides a cleaner answer which is simpler to interpret, especially for practical problems or further calculations.

Fraction Multiplication

Fraction multiplication involves multiplying the numerators and denominators separately. Unlike addition or subtraction, you don't need a common denominator for this operation. Set the fractions you wish to multiply, like 7/2 and 4/100. Multiply the numerators: 7 times 4 gives 28. Next, multiply the denominators: 2 times 100 gives 200. So, the multiplied fraction is 28/200. After finding the product, don’t forget to simplify it as we discussed earlier. Always check if there’s a common divisor, then simplify to find the neatest, most understandable form of your answer.