Understanding Decimals and Base 10 Fractions
Decimals and fractions often show up in real-life situations, especially when we’re dealing with money (like pounds and pence) and measurements (such as meters, centimeters, and millimeters). Learning to convert fractions to decimals, especially those with base 10 denominators (10, 100, 1000), makes these conversions straightforward and helps us work easily with small parts of a whole. Let’s dive into how decimals and fractions work together!
Base 10 Fractions and Decimal Places
We often encounter fractions like 1⁄10, 1⁄100, and 1⁄1000, called “base 10 fractions” because they use powers of 10 in their denominators. These base 10 fractions correspond directly to the decimal system. For example, 1⁄10 is equivalent to 0.1 because it places the digit 1 in the tenths place. Similarly, 1⁄100 becomes 0.01, and 1⁄1000 becomes 0.001. This makes it easy to see and write fractions with a denominator of 10, 100, or 1000 as decimal numbers.
If we have 2 tenths, for instance, we place a 2 in the tenths place to make 0.2. Counting up in tenths is straightforward: 0.1, 0.2, 0.3, all the way to 1.0. Once we reach ten-tenths, we complete one whole, moving to the ones place, giving us the number 1. Similarly, if we have 11 hundredths, we write this as 0.11, where 1 is in the tenths place and the other 1 in the hundredths place.
Decimals and Money: Pounds and Pence
Decimals are super helpful with money. Since 100 pence make a pound, we can think of each penny as a hundredth of a pound. So, 25 pence is like saying 0.25 pounds, 75 pence becomes 0.75 pounds, and so on. When we write £4 and 75 pence as £4.75, we’re simply placing the 75 in the hundredths (pence) part of the number, showing both pounds and pence as a decimal.
Similarly, if we know a price in pounds and want to convert it to pence, we multiply by 100. For example, £2.40 multiplied by 100 equals 240 pence. So, moving between pounds and pence using decimals and whole numbers makes it easy to handle money amounts.
Measuring Length
In the metric system, units of length are organized by powers of ten, making conversions straightforward. The millimeter (mm) is the smallest unit, with 10 millimeters equaling 1 centimeter (cm). A meter (m), the standard unit of length, contains 1,000 millimeters or 100 centimeters. For longer distances, the kilometer (km) is used, equivalent to 1,000 meters, 100,000 centimeters, or 1,000,000 millimeters. This decimal-based structure simplifies conversions by allowing movement of the decimal point.
Decimals and Metric Measurements
The metric system also works well with decimals, particularly when we’re dealing with lengths like meters, centimeters, and millimeters. Each meter has 100 centimeters and 1000 millimeters, so when we convert between these units, we can use decimals to represent smaller parts of a meter.
If we have 1.23 meters, for instance, this means we have 1 meter and 23 centimeters, as the 23 falls in the hundredths place (since there are 100 centimeters in a meter). Similarly, 0.9 kilometers equals 900 meters because each kilometer contains 1000 meters, and moving the decimal one place to the right gives us the result in meters.
Decimal Places and Placeholders
The decimal point separates whole numbers from fractions. Each place to the right of the decimal point represents tenths, hundredths, thousandths, and so on. If we have “extra” space, we use zeros as placeholders. For example, writing 0.07 means we have 7 hundredths, so we place a 0 in the tenths place to ensure the 7 is correctly aligned in the hundredths place. This way, we don’t accidentally write it as 7 tenths (0.7).
Converting Fractions to Decimals
To convert fractions to decimals, place the top number (numerator) in the appropriate decimal place based on the denominator. For instance, if we have 45⁄100, we place 45 in the tenths and hundredths places to get 0.45. Similarly, 3⁄10 becomes 0.3, while 125⁄1000 becomes 0.125, where 125 is placed in the thousandths place.
If we start with a decimal and want to convert to a fraction, we can also look at the smallest decimal place. For example, 0.29 translates to 29⁄100 since the last digit, 9, is in the hundredths place.
