Chapter Two — Rational Numbers
The Humongous Book of Algebra Problems
18
Rational Number Notation
Proper and improper fractions, decimals, and mixed numbers
2.1 Express 0.013 as a percentage.
Decimal numbers, like percentages, express a value in terms of a whole.
This whole value can be expressed in decimal form, as 1 (or 1.0), and as a
percentage, 100%. Transforming a decimal into a percent is as simple as moving
the decimal point exactly two digits to the right. Here, 0.013 = 1.3%.
2.2 Express 0.25% as a decimal.
According to Problem 2.1, converting from a decimal to a percentage requires
you to move the decimal point two digits to the right. It comes as no surprise,
then, that performing the opposite conversion, from percentage to decimal,
requires you to move the decimal exactly two digits to the left.
In this problem, only one digit, 0, appears to the left of the decimal. The
second, unwritten, digit is also 0. Therefore, 0.25% = 00.25% = 0.0025%.
Note: Problems 2.3–2.4 refer to the rational number .
2.3 Express the fraction as a decimal.
The rational number represents the quotient . To express the fraction as
a decimal, use long division to divide 4 by 1. Set up the long division problem,
writing an additional zero at the end of the dividend. Copy the decimal point
above the division symbol.
For the moment, ignore the decimal point within the dividend and imagine
that 1.0 is equal to 10. Because 4 divides into 10 two times, place a 2 above
the rightmost digit of 10. Because 4 does not divide evenly into 10, a remainder
will exist.
Multiply 2 in the quotient by the divisor (4) and write the result
below the dividend (10). Draw a horizontal line beneath 8.
You can put
as many zeroes
as you want at the
beginning of a decimal:
0.25, 00.25, 000.25,
and 0000000000.25 all
mean the same thing.
You can also add
zeroes at the end of
a decimal: 1.5, 1.50,
1.500, and so on.
The number
you’re dividing BY
is called the “divisor”
and the number you’re
dividing INTO is called
the “dividend.” The
answer you get
once you’re done
dividing is called
the “quotient.”
You can
add as many
zeroes as you want,
and you can do it at any
time during the problem.
Here’s your goal: you want
the answer to have either
terminated or begun to
repeat. If it hasn’t done
either, pop some more
zeroes up there and
keep going.
Chapter Two — Rational Numbers
The Humongous Book of Algebra Problems
19
Subtract 8 from 10 and write the result below the horizontal line.
The difference (10 – 8 = 2) is not 0, so the quotient has not yet terminated. Place
another 0 on the end of the dividend and on the end of the number below the
horizontal line.
This time, dividing the bottommost number by the divisor produces no
remainder; 4 divides evenly into 20. Write the result above the division symbol
next to the 2 already there.
Multiply the newest digit in the quotient (5) by the divisor (4) and write the
result beneath 20.
Subtract the bottom two numbers.
Because the remainder is 0, the division problem is complete:
.
Rational
numbers either
repeat or terminate.
When you get a
remainder of 0, you
stop dividing, and
the decimal
terminates.
Chapter Two — Rational Numbers
The Humongous Book of Algebra Problems
20
Note: Problems 2.3–2.4 refer to the rational number .
2.4 Express the fraction as a percentage.
According to Problem 2.3, . To transform a decimal into a percentage,
move the decimal point two digits to the right: 0.25 = 25.0%, or simply 25%.
Note: Problems 2.5–2.7 refer to the rational number .
2.5 Express the fraction as a decimal.
Use the method described in Problem 2.3 to rewrite the fraction as a long di-
vision problem. Here, however, there is no immediate need to place a 0 in the
dividend. No decimal point is written explicitly, so write one at the end of the
dividend and copy it above the division symbol.
Six divides into eleven one time, so write 1 above the rightmost digit of 11.
Multiply the divisor by the digit just written above the dividend , write
the result below the dividend and subtract.
The divisor (6) cannot divide into the result (5) a whole number of times. As
Problem 2.3 directed, change 5 into 50 and also add a zero to the end of the
dividend.
Six divides into 50 eight times, so place 8 at the right end of the quotient.
Multiply the new digit by the divisor (6), write the result below 50,
and subtract.
In Problem
2.3, you had
to add a zero
because 4 divides
into 10 but it really
doesn’t divide into 1
very well. In Problem
2.5, however, 6 does
divide into 11, so
you don’t need
to write 11 as
110.
When you’re
long dividing,
what you’re
dividing INTO has to
be bigger than what
you’re dividing BY. If
it’s not, add a zero
to the dividend
and the bottom
number.
Chapter Two — Rational Numbers
The Humongous Book of Algebra Problems
21
Six does not divide into 2 a whole number of times, so once again insert zeroes
after 2 and 11.0.
Six divides into 20 three times. Take the appropriate actions in the long division
problem.
Once again, 2 is the bottommost number in the long division problem. If you
place zeros after it and after the dividend, it produces another 3 in the quotient.
Once again, 2 is the bottommost number. Repeating this process is futile—each
time a zero is added, it produces another 3 in the quotient and a difference of
2, the same number with which you started. Because the division problem has
turned into an infinite loop producing the same pattern of digits, you can
conclude that .
Put a 3
above the
division symbol,
multiply it by the
divisor (6), write
that multiplication
result (18) below
20, and then
subtract it
from 20.
If the
decimal form of
a rational number
doesn’t terminate,
one or more digits will
repeat innitely. Write
the decimal with a
little bar over the
3 to indicate that
it’s an innitely
repeating digit in
the decimal.
Chapter Two — Rational Numbers
The Humongous Book of Algebra Problems
22
Note: Problems 2.5–2.7 refer to the rational number .
2.6 Express the fraction as a percentage.
According to problem 2.5, . Move the decimal two places to the
right to convert the decimal into a percentage: .
Note: Problems 2.5–2.7 refer to the rational number .
2.7 Express the fraction as a mixed number.
The fraction is considered improper because its numerator is greater than
its denominator. To express it as a mixed number, divide 11 by 6. There is no
need to use long division—it is sufficient to conclude that 6 divides into 11 one
time with a remainder of 5. Thus . The whole number portion of the
mixed number is the number of times the divisor divides into the dividend; the
remainder is the numerator of the fraction. The denominator of the fraction
matches the denominator of the original fraction.
In other words, given the improper fraction , if x divides into y a total of w
times with a remainder of r, then .
2.8 Express the improper fraction as a mixed number.
Four divides into 65 a total of 16 times with a remainder of 1. Therefore
. The fractional part of the mixed number consists of the remainder
divided by the original denominator.
2.9 Express as an improper fraction.
To convert a mixed number into a fraction, multiply the denominator and the
whole number and then add the numerator. Divide by the denominator
of the mixed number: .
2.10 Express as an improper fraction.
According to Problem 2.9, . Substitute a = 4, b = 5, and c = 12 into
the formula.
Improper
doesn’t mean
unacceptable.
It’s ne to have
fractions with bigger
numbers on top than
on bottom, and most
teachers would
rather you leave
fractions in improper
form, rather than
write them as
mixed numbers.
Here’s a
quick way to
gure this out.
Either long divide
or use a
calculator. You get
16.25. The number left
of the decimal (16) is
the nonfraction part
of the mixed number.
To gure out the
remainder, multiply
that whole number
by the denominator
and
subtract what you
get from the
numerator:
65 – 64 = 1.
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