|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Unit 3.Operating with fractions |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Look through the table and try to memorize it. |
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Examples |
Rules |
|
|
|
|
written |
read |
|||||||||||||||
|
|
|
|
|
||||||||||||||||
|
|
|
|
|
|
|
1/2 |
|
|
|
|
|
One second; or: a half |
½; 1/3; ¼; 2/3; 1/100; and 5/16 are |
||||||
|
|
|
|
|
|
|
1/3 |
|
|
|
|
|
one third; a third |
|||||||
|
|
|
|
|
|
|
2/3 |
|
|
|
|
|
Two thirds |
proper fractions. |
||||||
|
|
|
|
|
|
|
1/4 |
|
|
|
|
|
One fourth; or: a fourth; |
A proper fraction is one whose |
||||||
|
|
|
|
|
|
|
|
|
|
|
|
or: a quarter |
numerator is less than denominator |
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
1/100 |
|
|
|
A hundredth |
|
|||||||||||
|
|
|
|
5/16 |
|
|
|
|
Five sixteenths |
|
||||||||||
|
|
|
|
23/6 |
|
|
|
|
Twenty three six |
23/6 and 9/9 are improper fractions. An |
||||||||||
|
|
|
|
|
|
|
|
improper fraction is a fraction, whose |
||||||||||||
|
|
|
|
|
|
|
9/9 |
|
|
|
|
|
Nine ninths |
numerator is equal to or larger than the |
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
denominator |
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
Three and five seventh is a mixed |
||
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
Three and five sevenths |
number. A mixed number is a number |
|||||
|
|
|
|
|
|
|
7 |
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
and a fraction written together |
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ar over br equals a over |
To reduce a fraction to its lowest terms, |
|
|
|
|
|
ar |
= |
a |
divide the numerator and the |
||||||||||||
|
|
|
|
|
br |
|
b |
|
|
|
b |
denominator by their highest common |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
factor (or: measure, or: divisor) |
|
|
|
|
|
a |
|
|
|
|
|
ar |
|
|
|
a over b equals ar over |
To reduce a fraction to higher terms, |
||||
|
|
|
|
|
= |
|
|
|
|
br |
multiply the numerator and the |
|||||||||
|
|
|
|
|
b |
br |
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
denominator by the same number. |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a over b, this fraction |
To find the sum (the difference) of two |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
followed by plus or |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
unlike fractions, change them to like |
|
|
a |
|
|
c |
|
|
|
|
|
ad ±bc |
|
minus c over d equals ad |
||||||||
|
± |
= |
|
fractions (fractions having their least |
||||||||||||||||
|
b |
d |
|
|
|
|
|
bd |
plus or minus bc this |
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
common denominator) and combine the |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
sum or difference over |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
numerators. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
bd |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a |
|
|
|
|
c |
|
|
|
|
|
ac |
|
|
a over b, this fraction |
To find the product of two fractions, |
||
|
|
|
× |
|
= |
|
multiplied by c over d |
multiply the numerators together and the |
||||||||||||
|
|
|
b |
d |
|
bd |
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
equals ac over bd |
denominators together |
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
9
|
5 |
|
|
1 |
|
|
|
5 |
|
|
|
2 |
|
|
Five sixths divided by a |
|
To find quotient of two fractions, |
|||||
* |
|
= |
|
=1 |
|
|
half equals one and two |
|
multiply the dividend by the inverted |
|||||||||||||
6 |
|
|
|
3 |
|
|
|
|||||||||||||||
2 |
|
|
|
|
3 |
|
|
thirds |
|
|
divisor |
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a over b, this fraction |
|
To convert an improper fraction into a |
||
|
|
|
a |
|
|
|
c |
|
|
|
|
ad |
|
|
|
mixed number, break it up into the sum |
||||||
|
|
|
/ |
|
|
= |
|
|
divided by c over d |
|
||||||||||||
|
|
|
b |
|
d |
|
|
bc |
|
|
of an integer and a proper fraction: |
|||||||||||
|
|
|
|
|
|
|
|
|
|
equals ad over bc |
|
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
|
|
3.1Assignments |
|
|
|
|
||||||||||
3.1.1. Memorize the following words and word-groups: |
||||||||||||||||||||||
proper fraction |
|
[`prOpq `frxkS(q)n] |
правильная дробь |
|||||||||||||||||||
improper fraction |
|
[ Im`prOpq frxkS(q)n] |
неправильная дробь |
|||||||||||||||||||
mixed fraction |
|
[mIkst`frxkS(q)n] |
смешанная дробь |
|||||||||||||||||||
to reduce a fraction |
[rI`djHs q `frxkS(q)n] |
Приводить дробь |
||||||||||||||||||||
a) to its lowest terms |
[` louIst `tq:mz] |
к наименьшему |
||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
[`haIq ` tq:mz] |
|
значению |
|
b) to its higher terms |
|
к наибольшему |
||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
[`kOmqn` fxktq] |
значению |
||
common factor / measure / |
общий делитель |
|||||||||||||||||||||
divisor |
|
|
|
|
|
|
|
|
|
|
[`meZq] [dI`vaIzq] |
|
||||||||||
combine |
|
|
|
|
|
|
|
[kOm`baIn] |
|
сочетать |
||||||||||||
to convert |
|
[kOn`vq:t] |
|
преобразовывать |
||||||||||||||||||
an integer |
|
[`IntIGq] |
|
целое число |
||||||||||||||||||
integer solution |
|
|
|
|
решение в целых |
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
числах, целочисленное |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
решение |
complex / Gaussian integer |
|
|
|
Комплексное целое |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
[`reISIou] |
|
число |
|
ratio |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
отношение, |
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
коэффициент, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
пропорция, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
[[`In`vWs `reISIou] |
соотношение |
||
inverse ratio |
|
отношение обратных |
||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
величин |
3.1.2.Give the definitions:
•a proper fractions
•an improper fractions
•a mixed fractions
3.1.3.Do these operations:
•reduce a fraction to higher terms: 23
10
•reduce a fraction to its lowest terms: 128
•find the sum of two unlike fractions: 23 + 34
•find the product of two fractions: 53 ×125
•convert an improper fraction into a mixed number: 248 ÷ 99
•find quotient of two fractions : 456 *122
3.1.4.Match the columns:
1 |
123/123 |
a. |
A proper fractions |
||
2 |
23/9 |
b. |
a mixed fractions |
||
3 |
1/8 |
c. |
Braces |
||
4 |
2 |
2 |
|
d. |
An improper fractions |
|
5 |
|
|
|
|
5 |
= |
|
|
e. |
Less than |
6 |
{ } |
|
f. |
a mixed fractions |
|
7 |
< |
|
|
g. |
is equal to, or: does not equal |
8 |
Θ |
|
|
h. |
Infinity |
9 |
∞ |
|
|
i. |
Because |
3.1.5.Read these operations:
•55ba = ba
•1 ± 1 = 1* 2 ± 2 *1 2 2 * 22
•ba = 66ba
•3 × 3 = 3*3
7 5 7 *5
•7 / 7 = 7 *5
5 5 5* 7
•56 * 12 = 53 =1 23
3.1.6. Write these operations:
•a over b, this fraction followed by plus or minus c over d equals ad plus or minus bc this sum or difference over bd
•a over b, this fraction divided by c over d equals ad over bc
•ar over br equals a over b
•Five sixths divided by a half equals one and two thirds
•a over b, this fraction multiplied by c over d equals ac over bd
•a over b equals 2a over 3b
11