[MUSIC PLAYING] Let's look at our objectives for today. We'll start by thinking about numbers both as decimals and fractions. We'll then look to see how we can use decimal place values to convert a decimal into a fraction. We'll see how we can use the greatest common factor to simplify a fraction. And finally, we'll look to see how we can use calculators to check our work to convert a fraction back into a decimal.
Let's start by looking at what kind of numbers can be written both as a decimal and as a fraction. Many of the numbers that we use in mathematics can be written as both a fraction and a decimal. These numbers are called rational numbers. Rational numbers have a fraction and a decimal form which are equivalent. Converting between decimals and fractions is very useful when working with measurements, as well as everyday situations such as percentages.
Let's look at an example of how we use decimals and fractions with measurement. If we look at this ruler, which is measuring in centimeters, we can see 3.5 centimeters would be here on the ruler. We can also think of 3.5 centimeters as a fraction, 3 and 1/2 centimeters. And we can see that by looking at our tick marks. Here, the tick mark is halfway between the 3 and the 4, so 3 and 1/2 centimeters.
In order to convert a decimal to a fraction, we use the concept of decimal place value. Place value refers to where a number is located in relation to the decimal point. In the diagram below, the number 0.1234 shows the 1 that in the tenths place, the 2 in the hundredths place, the 3 in the thousandths place, and the 4 in the ten-thousandths place.
The fraction equivalent of a decimal is determined by the digits and their decimal place value. So for example, 0.267 is equivalent to the fraction 267/1000, because the last digit, 7, is in the thousandths place. Let's look at a few more examples.
In the decimal 0.3, 3 is in the tenths spot, so the fraction is 3/10. 0.51 has 1 in the hundredths spot, so the fraction is 51/100. For 0.223, the 3 in the thousandths place, so the fraction is 223/1000. And for the decimal is 0.8231, the 1 is in the ten-thousandths place, so the fraction is 8310/10000. Notice that our denominators are always a power of 10, which is a 1 followed by some number of zeros.
Now let's look at an example of converting a decimal into a fraction that will need some simplification. We want to convert the decimal 0.34 into a fraction. To determine the denominator of our fraction, we look at the righter-most nonzero digit of the decimal. So here, the 4 is in our hundredths place. So 0.34 as a fraction is 34/100.
We can simplify this fraction by looking for the greatest common factor of the numerator and the denominator, or the largest integer that divides both the numerator and the denominator. The factors of 34 are 1, 2, 17, and 34. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The common factors are 1 and 2, so the greatest, or largest, common factor is 2.
The shared factors can be canceled out by dividing both the numerator and denominator by the greatest common factor. Therefore, when we simplify 34/100, we divide 34 by 2 and 100 by 2, which gives us 17/50. Any common factor can be used to reduce a fraction, but further simplification will be necessary. However, using the greatest common factor will always produce a fraction in simplest form. So we know this fraction is in simplest form because the greatest common factor of the numerator and denominator is now 1.
Let's look at another example. Here we want to convert 1.205 to a fraction. The last digit, 5, is in the thousandths spot, so the fraction equivalent is 1205/1000. The greatest common factor of 1,205 and 1,000 is 5. Therefore, we can simplify by dividing both the numerator and denominator by 5, or canceling out the 5 in the numerator and in the denominator. So 1205/1000 becomes 241/200.
Finally, let's see how we can use a calculator to convert our fraction back into a decimal to make sure that we have correctly converted our decimal into a fraction. All fractions are equivalent to dividing the numerator by the denominator. So the fraction 3/4 is the same as 3 divided by 4. In the calculator, you simply press 3, the division sign, 4, and then Enter.
Let's look at our previous example. We converted 1.205 to the fraction 241/200. Moving backwards to check our answer, we know that 241/200 is the same as 241 divided by 200. Therefore, in a calculator you simply press 241, press the division sign, then press 200, and then Enter or equal sign, and you will see that the decimal equivalent is indeed 1.205.
Let's look at our important points from today. Make sure you get them in your notes so you can refer to them later. Rational numbers can be written as fractions and decimals. To convert a decimal to a fraction, use decimal place value and the digits of the decimal form. We can simplify fractions by canceling out or dividing the numerator and denominator by their greatest common factor. And a fraction can be converted to a decimal using division on a calculator.
So I hope that these key points and examples helped you understand a little bit more about converting a decimal to a fraction. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
The largest positive number that divides both a and b.