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ALGEBRA
- solved problems |
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Adding and subtracting decimals |
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48. |
Add and subtract given decimals,
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Solutions: |
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a)
7.9 + 12 + 0.147 =
20.047,
b)
215.04 + 3.756 + 17.3 = 236.096 |
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7.900
215.040 |
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12.000
3.756 |
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+
0.147
+ 17.300 |
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20.047
236.096 |
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c)
41.56
− 9.178 =
32.382,
d)
5.08
− 0.9937 = 4.0863 |
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41.560
5.0800 |
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−
9.178
−
0.9937 |
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32.382
4.0863 |
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To add or subtract decimals, insert zeros in empty decimal place
values so that all of the numbers have the same number of decimal
places. |
Write the corresponding decimal places one under another,
i.e., the decimal points should be vertically aligned, and then follow
the rules for adding or subtracting whole numbers. |
Line up the
decimal point in the answer. |
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Multiplying and dividing decimals
by powers of ten or by decimal units
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49. |
Multiply and divide given decimals by powers of
ten.
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Solutions: |
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a)
0.58 ·10 =
5.8,
b)
79.4 ·100 =
7940,
c)
0.001 ·10000 = 10, |
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d)
30.07 ¸
10 = 3.007,
e)
0.08 ¸
100 = 0.0008,
f)
12.0 ¸
1000 = 0.012. |
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To multiply decimals by powers of
10, such as 10, 100, 1000 etc. move the decimal point right as many places as there are
zeros in the power. |
To divide decimals by powers of
10, move the decimal point left as many places as there are zeros in the power.
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50. |
Multiply and divide given decimals by decimal
units.
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Solutions: |
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a)
83.05 · 0.1 = 8.305, b)
547.2 · 0.001 = 0.5472, c)
0.9 · 0.0001 = 0.00009, |
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d)
0.75 ¸
0.001 = 750,
e)
4.035 ¸
0.01 = 403.5,
f)
29.0 ¸
0.1 = 290. |
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To
multiply decimal by the decimal
units; 0.1, 0.01, 0.001, ... Move the decimal point of the decimal left as many places as there are
zeros in the decimal unit. |
Divide decimal by the decimal units by
moving the decimal point of the decimal right as many places as there are zeros in the decimal unit. |
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Multiplication and division of decimal numbers
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51. |
Multiply given decimals.
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Solutions: |
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a)
0.589 · 47.8,
b)
79.6 · 0.00503,
c)
629.7 · 1.03 |
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2356 3980
6297 |
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4123
000
0000 |
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4712
2388
18891 |
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28.1542
0.400388
648.591 |
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To multiply decimal numbers, ignore the decimal points and multiply
the digits. Count the total number of decimal places in both decimal
numbers being multiplied.
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Place a decimal point in the answer so
that it has as many digits to the right of the decimal point as the total
number of decimal places in the two decimals being multiplied. |
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52. |
Divide given decimal numbers.
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Solutions: |
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a)
467.5 ¸ 17 = 27.5,
b)
2.773 ¸ 0.47 = (2.773 ·
100) ¸
(0.47 · 100) = |
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−
34
= 277.3 ¸
47 = 5.9 |
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127
− 235 |
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− 119
423 |
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85
− 423 |
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− 85
0 |
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0 |
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c)
0.15059 ¸ 3.7 = 1.5059 ¸
37 = 0.0407, d)
10.8 ¸ 0.004 =
10800 ¸
4
=
2700 |
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150
− 8 |
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− 148
28 |
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259
− 28 |
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− 259
0 |
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0 |
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To divide decimal numbers, first make the divisor into a whole
number by multiplying both the dividend and the
divisor by the same
power of 10 (such as 10, 100, 1000, . . .) or move a decimal point in
the divisor to the right end, and in the dividend the same number of
places. |
To divide a decimal by a whole number, divide as you would
for whole numbers. Place a decimal point in the answer so that it
lines up with the decimal point in the dividend.
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If after dividing you
have a remainder, add a zero to the dividend and continue to divide
until there is no remainder or the decimals repeat. |
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Rational numbers
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53. |
Compare given rational numbers.
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Solutions: |
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since
(-4)
· (-10)
= 5
· 8 |
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since
3 · 35 = (-7)
· (-15) |
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since
5 · 7 = 8 · 4 |
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since
(-9)
· 3 < 5
· (-5) |
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Equivalent rational numbers or fractions
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54. |
Write equivalent rational numbers or fractions.
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Solutions: |
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Convert a rational number or fraction to an equivalent fraction by multiplying the numerator and denominator
by the same nonzero
number. Equivalent
fractions are different fractions that represent the same
number. |
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Reducing
or simplifying rational numbers or fractions |
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55. |
Reduce or simplify given rational numbers or
fractions.
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Solutions: |
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To reduce or simplify a fraction to lowest terms, divide the
numerator and denominator by their greatest common divisor (gcd)
or (greatest common factor - gcf). |
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Adding and subtracting rational
numbers or fractions
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56. |
Add and subtract rational numbers or fractions.
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Solutions: |
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12 = 2 · 2 · 3, 36 = 2 ·
2 · 3 · 3, 48 = 2 · 2 ·
2 · 2 · 3 |
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LCD(12, 36, 48) = 2 · 2 · 2 · 2 · 3 · 3 = 144 |
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Find the least common denominator, write equivalent fractions, then
add or subtract the fractions. Reduce if necessary. |
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Multiplication of
rational numbers
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57. |
Multiply rational numbers or fractions.
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Solutions: |
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Rational numbers are multiplied by multiplying
numerators and multiplying denominators. Change any mixed numbers to improper fractions. Reduce fractions before
multiplication. |
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Division of
rational numbers, reciprocal
fractions (multiplicative inverse) |
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58. |
Divide rational numbers or fractions.
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Solutions: |
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Dividing a rational number or fraction by another
fraction is equivalent to multiplying the dividend with the reciprocal of the divisor.
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Simplifying complex or compound
fractions
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59. |
Simplify rational numbers or fractions.
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Solutions: |
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A complex or compound fraction is a fraction whose numerator
and/or denominator are also a fraction or mixed number.
Divide the numerator by the denominator.
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