HP 9g User manual

hp calculators
HP 9g Solving Problems Involving Fractions
Basic Concepts
Fractions on the HP 9g
Practice Working Problems Involving Fractions

hp calculators
HP 9g Solving Problems Involving Fractions
Basic concepts
Those numbers that can be written as one integer over another, i.e. b
a, are called rational numbers. Note that b can’t be
zero. When written as the quotient of two integers, rational numbers are called fractions. In arithmetic there are three
basic rules for fractions:
♦d
c
b
a>if ad – bc > 0 (same for < and =)
♦bdbcad
d
c
b
a
+
=+
♦bd
ac
d
c
b
a=×
b
ais also referred to as a vulgar fraction when aand bare positive integers (the signis considered apart) .ais called
the numerator (corresponds to the dividend in a division) and bis the denominator (which corresponds to the divisor in
a division). When the numerator is 1 (or –1), it is a unit fraction. A proper fraction is a fraction in which the numerator
(apart from the sign, remember) is less than the denominator. Therefore, proper fractions always lie between –1 and 1. If
the numerator is greater than the denominator, the fraction is called improper.
Those vulgar fractions that have the same value are called equivalent fractions: for example 4
3and 8
6. Reducing a
vulgar fraction to its lowest terms means to find the simplest equivalent fraction, which can be done by dividing the
numerator and the denominator by the same number. This process is also called cancellation.
Mixed numbers are those improper fractions written as an integer followed by a proper fraction. For example: 4¾ or 2½.
It is important to understand that there’s no implicit multiplication in c
b
a(also written as c
b
a). In fact, it is an addition
that is implicit:
cbac c
b
aa c
b
+
=+=
When the numerator and the numerator of a fraction are not both integers then the fraction is called complex, for
example:
4
3
3
13 . Finally, a number where the part which is a proper fraction is expressed as a set of digits placed after a
decimal point, is called a decimal (also known as decimal fraction) e.g. 50
7
3= 3.14.
Fractions on the HP 9g
The HP 9g has three keys to handle fractions, namely O, ~o and ~n . The symbol used by the HP
9g to show a fraction (i.e. the equivalent to the symbol “/”, sometimes called solidus) is “ ” and is entered into the entry
line by pressing O. Thus, 7 8 means 87 and is entered by pressing 7O8. Mixed numbers are also
keyed in using the Okey twice, for instance: 7 89 means 9
8
7and is entered into the entry line by pressing 7
O8O9 but this number is displayed in the result line (i.e. once =has been pressed) as 7 89.
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hp calculators
HP 9g Solving Problems Involving Fractions
~o is used for converting a mixed number to an improper fraction and vice versa, and ~n for converting
a decimal to and from a fraction. Let’s illustrate all this with various examples.
Practice working problems involving fractions
Example 1: Enter the proper fractions 9
3and 124 21
Solution: Note that pressing 3/9= does not return a fraction but a decimal number which is the result—
within the accuracy of the calculator—of dividing 3 by 9. As stated above, fractions are entered with the
Okey, which separates the numerator from the denominator. Press:
3O9
The entry line now reads 3 9. Press =to display the result line. The fraction now displayed is 1 3,
which is equivalent to the entered fraction but reduced to its simple form. The HP 9g always tries to find
the simplest equivalent fraction. Let’s enter the second fraction by pressing:
21O124=
No reduction is possible this time, therefore the fraction displayed in the result line is 21 124.
Example 2: Enter the improper fraction 101
1000 and the complex fraction 981.
Solution: These fractions are entered exactly as in the previous example, but the results displayed after pressing the
=key are different. Let’s enter the first fraction by pressing:
1,3O101=
The fraction displayed in the entry line (1E3 101) changes to 9 91 101 which means 101
91
9. Improper
fractions are always converted to mixed numbers (i.e. an integer plus a proper fraction) after pressing =.
As with proper fractions, the HP 9g tries to give the simplest form. As to the complex fraction, press:
1.8O9=
It returns 0.2 and not the fraction 5
1. This is because calculations containing both fractions and decimals
are calculated in decimal format. As long as both the numerator and the denominator evaluate to an
integer, the result will be in fraction format, though. For example √(4) 3 will return 2 3.
Example 3: Enter the mixed numbers 18
2
7, 125
57
1−and 5
19
3.
Solution: When entering mixed numbers, remember that the Okey is used for separating both the integer from
the proper fraction and the numerator from the denominator. To enter the first fraction press:
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HP 9g Solving Problems Involving Fractions
7O2O18=
The fraction returned is 7 19, i.e. 9
1
7which is the same as 18
2
7after doing a cancellation.
Let’s now enter the second fraction 125
57
1−. On the HP 9g negative fractions and mixed numbers are
keyed in by pressing the M key before entering any part (integer, numerator or denominator). It is
usually pressed just before the integer of a mixed number or the numerator of a fraction. In this example
press:
M1O57O125=
No cancellation is possible this time, so the result is –1 57 125.
Note that the third number is not actually a mixed number strictly speaking because its fraction part is not
proper. Nevertheless, your HP 9g can handle it as well and will return the reduced, proper form. Press:
3O19O5=
which returns the mixed number 5
4
6.
Fractions can be entered and displayed wherever ordinary decimal numbers can be used. For example fractions can be
used in ordinary arithmetic, in calculations with logarithmic and trigonometric functions and also in programs. But
fractions cannot beused in Base-N mode – operations in this mode only work with integers.
Example 4: Add 4
3
1to 8
5
2
Solution: We will enter the two mixed numbers as explained above. No parenthesis is necessary because fractions
(“ ”) take priority over the addition. Press:
1O3O4+2O5O8
The entry line now reads 1 34 + 2 58. Press =to carry out the calculation.
Answer: The result line reads 4 38 which means 8
3
4. Since this calculation only contains fractions then the
result is expressed as a fraction too.
Example 5: Express the previous result as a decimal.
Solution: F↔D (~n) is a one-argument function, aF↔D, that converts a decimal to a fraction or vice versa.
Since the result of the previous example is stored in ANS, we don’t need to enter it again, just press:
~n=
Answer: 4.375. Since pressing the =key repeats the most recent calculation, if you press =now, the result line
will contain the mixed number again. Press =one more time to display the decimal fraction.
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HP 9g Solving Problems Involving Fractions
Example 6: Are the fractions 489
147 and 1793
539 equivalent to each other?
Solution: There are several ways of testing whether two fractions are equivalent. We can subtract one from the other
and see if the result is zero, or we can calculate the cross multiplication since if a/b = c/d then ad = bc.
Doing the division is another way. But let your HP 9g do the hard part, and just enter both fractions. Since
the Hp 9g carries out an automatic simplification, if the fractions displayed in the result line are the same
then they are equivalent. Press:
147O489= which returns 49 163
539O1793=
Answer: Since the latter results in 49 163 too, they are equivalent fractions.
Example 7: Convert the mixed number 133
89
2to an improper fraction.
Solution: We have seen how improper fractions are automatically converted to mixed numbers when evaluated. But
the opposite is also possible. The e
d
c
b
A
↔
function (~o ) carries out conversions between
mixed numbers and improper fractions. Press:
2O89O133~o=
Answer: 355 133, i.e. 133
355 . Incidentally, this is a good approximation to π(within 8.47 millionths of one percent)
and very easy to remember because it’s made by duplicating the first three odd numbers and inserting a
division sign in the middle 355133 . And it was also the very first example in the HP-35 operating
manual!
Note. e
d
c
b
A↔and F↔D must be the last functions entered into the entry line, otherwise a syntax error occurs.
However, chain calculations are always possible using the ANS function. For example:
Example 8: Calculate 4
3
30 +)º(sin and express the result as a fraction.
Solution: The calculation in question is: (
+
)º(sin 30 3 4F↔D)e
d
c
b
A
↔
. Trouble is that F↔D cannot be
placed in the middle of a calculation. We have to split it this way:
H30~z=†+3O4~n=
We now have the mixed number 4
1
1in ANS. (The sequence ~z= inserts the symbol º after 30,
so as not to be affected by the current angle mode). We can now convert it to an improper fraction by
pressing:
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HP 9g Solving Problems Involving Fractions
~o=
Answer: 54, i.e. 4
5
Example 9: Find a fraction which approximates πto four decimal places.
Solution: First set the number of displayed decimal places to 4 by pressing ~Ü4. Notice that πF↔D will
not return a fraction, because πis calculated to 24 significant digits. Too many digits for F↔D to handle.
(Also, bear in mind that denominators on the HP 9g must be less than 10000).πhas to be rounded to four
decimal places before attempting the conversion. To do so use the RND command in the MATH menu:
Y3~Õ~n=
The mixed number 3 177 1250 is returned. If a fraction is preferred, convert this number to an improper
fraction by pressing:
~o=
Answer: 14163
1250
3927
31250
177 .==
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