I Used to Know That_ Stuff You Forgot From School - Caroline Taggart [18]
Write 4 at the top of the sum, next to the 1, and carry 44 forward into the next column to make 441. How many times does 63 go into 441? Well, 6 goes into 44 seven times (6 x 7 = 42), so let’s try that. And, conveniently, 63 x 7 = 441. Which means that 63 goes into 441 exactly seven times, with nothing left over, and that answers the problem: 147.
Fractions, Decimals, and Percentages
☞ PROPER FRACTIONS
A fraction is technically any form of number that is not a whole number; what most people think of as fractions—numbers such as ½, ⅔, ¾, and so on—are properly called vulgar, simple, or common fractions (as opposed to decimal fractions; see page 60).
The top number in these fractions is called the numerator, the bottom one the denominator (remember, denominator down).
In fact, the examples given above are all proper fractions, with the numerator smaller than the denominator (the fraction represents less than 1). In an improper fraction the reverse is true, as in (an approximation for pi, see page 73), which can also be written as , because 7 goes into 22 three times, with a remainder of 1.
If you want to solve problems that involve fractions, it is important to know that if you divide or multiply both the numerator and denominator by the same number, you produce a fraction that is the same value as the original fraction. Take ½. Multiply both numerator and denominator by 2 and you get Which is still a half, because 2 is half of 4. Or multiply ½ by 3 and you get . Which again is still a half, because 3 is half of 6.
The same principle applies to division: If you start with and divide top and bottom by 3, you reduce your fraction down to ½ again. This process is called canceling. When you can’t cancel anymore, the fraction is in its lowest terms.
With addition and subtraction, however, you can only add and subtract fractions that have the same denominator. You can add ½ + ½ and get , which equals 1, because two halves make a whole. But what you have done is add the two numerators together. The denominator stays the same, because you are adding like to like. (It’s no different from adding 1 apple to 1 apple to get 2 apples.)
Now say you want to add ½ + ⅓. It’s easy to do, but first you must convert them so they have the same denominator. The lowest common denominator of 2 and 3 (the smallest number into which both will divide) is 6. To turn ½ into sixths, you need to multiply both parts of the fraction by 3:
So ½ is the same thing as .
To convert ⅓ into sixths, you need to multiply both parts by 2:
So ⅓ is the same thing as .
Now you have something that you can add, on the same principle of adding the numerators together:
The same applies to subtraction:
But both 4 and 10 can be divided by 2, to give the simpler fraction ⅖.
☞ DECIMAL FRACTIONS
The word decimal refers to anything with the number 10, and the English system is based on multiples of 10. As previously mentioned in the multiplication section, a single-digit number—say, 6—means that you have six units of whatever it is. When you have more than nine, you have to use two digits, with one digit representing the tens on the left and one digit representing the units on the right.
Decimal fractions work on the same principle, except that they go from right to left. The fraction is separated from the whole number by a dot called a decimal point. The figure immediately to the right of it represents tenths, to the right of that is hundredths, and so on. So 1.1 (pronounced one point one) = 1 plus one tenth of 1; 1.2 = 1 + 2/10 (or ⅕); 1.25 (pronounced one point two five) , or .
An interesting example is 1.25, because it is the same as 1¼. How do we know that? Well, return to the idea of dividing numerators and denominators by the same thing. For example, can be