Math Olympiad 3

Week	Topic and action
Week 11	Fractions & Decimals (1) Practice Evaluation

Fraction number

What is fraction?

A fraction is a number that expresses part of a group.
Fractions are written in the form

or a/b, where a and b are whole numbers, and the number b is not 0.
The number a is called the numerator, and the number b is called the denominator.
The following are fractions of the corresponding shaded region as a fraction of the whole circle.

Comparing fraction
The value of a fraction is equal to its numerator divided by its denominator. If two fractions has the same value, they are equivalent fractions.
In the above fractions,
1
2
and
6
12
are equivalent fractions, while
2
3
and
4
6
are equivalent fractions.
When comparing two fractions, evaluating them to decimals and then doing comparison is the final resort. Usually, you can apply the following first:
1. To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator. For example, >
  2
  5
  .
2. To compare fractions with the same numerator, look at their denominators. The larger fraction is the one with the smaller denominator. For example, >
  3
  6
  .
3. To compare fractions with different numerators and denominators, take the cross product. The first cross-product is the product of the first numerator and the second denominator. The second cross-product is the product of the second numerator and the first denominator. Compare the cross products using the following rules:
  - If the cross-products are equal, the fractions are equivalent.
    1
    3
    =
    3
    9
    since 1 × 9 = 3 × 3.
  - If the first cross product is larger, the first fraction is larger.
    2
    3
    >
    3
    5
    since 2 × 5 > 3 × 3.
  - If the second cross product is larger, the second fraction is larger.
    2
    3
    <
    3
    4
    since 2 × 4 < 3 × 3.
Reduce fraction
A fraction is in lowest terms if the greatest common factor of its numerator and denominator is 1. The following fractions are in their lowest terms.

1
2
,
1
3
,
2
3
,
2
5
,
5
6
,
7
10
, ...
While the below fractions are not in their lowest terms.

2
4
,
3
6
,
4
8
,
7
21
, ...
When a fraction is not in its lowest terms, you can always reduce it to a lower terms by dividing a common factor (larger than 1) of its numerator and denominator until the fraction is in its lowest terms.
For example,
4
8
=
4 ÷ 2
8 ÷ 2
=
2
4
=
2 ÷ 2
4 ÷ 2
=
1
2
.
Improper Fractions
Improper fractions have numerators that are larger than or equal to their denominators.

5
4
,
7
7
, and
11
2
are examples of improper fractions.
Mixed Numbers
Mixed numbers have a whole number part and a fraction part.

1 2
3
( = 1 +
2
3
),
2 3
4
( = 2 +
3
4
), and
3 1
2
( = 3 +
1
2
) are examples of mixed numbers.
Converting Mixed Numbers to Improper Fractions
To change a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator of the fractional part.
For example:

1 2
3
=
1 × 3 + 2
3
=
5
3

3 1
2
=
3 × 2 + 1
2
=
7
2
Converting Improper Fractions to Mixed Numbers
To change an improper fraction into a mixed number, divide the numerator by the denominator. The quotient is the integer part and the remainder is the numerator of the fractional part.
For example,

13
5
=
2 3
5
(13 ÷ 5 = 2 ...R 3)

15
7
=
2 1
7
(15 ÷ 7 = 2 ...R 1)
Reciprocal
- The reciprocal of a fraction is obtained by switching its numerator and denominator. For example, the reciprocal of
  3
  4
  is
  4
  3
  .
- To find the reciprocal of a mixed number, first convert the mixed number to an improper fraction, then switch the numerator and denominator of the improper fraction. For example, to fine the reciprocal of
  1 3
  4
  , first to get its improper form,
  1 × 4 + 3
  4
  =
  7
  4
  , then the reciprocal is
  4
  7
  .

Decimal number

Decimal number: A fractional number that is written using base ten notation and expressed using a decimal point.
- The decimal number 0.1 is equivalent to the fraction
  1
  10
  . You read both the same way--one tenth. The period to the left of the 1 is called a decimal point. The first place to the right of the decimal point is called the tenths' place.
- The decimal number 0.02 is equivalent to the fraction
  2
  100
  . You read both the same way--two hundredth. The second place to the right of the decimal point is called the hundredths' place.
- The decimal number 0.003 is equivalent to the fraction
  3
  1000
  . You read both the same way--three thousandths. The third place to the right of the decimal point is called the thousandths' place.
- The decimal number 1.321 is equivalent to 1 plus the fraction
  321
  1000
  = 1 +
  3
  10
  +
  2
  100
  +
  1
  1000
  . You read both the same way--one and three hundred twenty-one thousandth. Remember to put "and" between the whole number part and the fraction part, just as in a mixed number.
Comparing decimals
When decimals are compared, you should start with their greatest place values.
- Example 1. Compare 8.87 and 9.77. First compare the ones: 8 < 9, so 8.87 < 9.77.
- Example 2. Compare 1.21 and 1.19. First compare the ones: 1 = 1. Next compare the tenths: 0.2 > 0.1, so 1.21 > 1.19.
- Example 3. Compare 0.341 and 0.3399. First compare the ones: 0 = 0. Second compare the tenths: 0.3 = 0.3. Third compare the hundredths: 0.04 > 0.03, so 0.341 > 0.339.0.3399
Decimals as Fractions
Decimals can be written as fractions, and fraction can be written as decimals.
- Example 1.
  67
  100
  can be written as 0.67.
- Example 2. 0.38 can be written as
  38
  100
  .
Comparing decimals and Fractions
- Example 1. compare 1 +
  3
  4
  and 1.56. First you can write the mixed number 1 +
  3
  4
  as a decimal:1.75. Now compare 1.75 and 1.56. Compare ones:1 = 1. Compare the tenths:0.7 > 0.5. So 1.75 > 1.56.
- Example 2. compare
  1
  5
  and 0.26. First you can write the fraction
  1
  5
  as a decimal 0.2. And you can write 0.2 as 0.20, then you can compare 0.20 and 0.26. Compare the ones:0 = 0. Compare the tenths:0.2 =0.2. Compare the hundredths:0.00 < 0.06. So 0.20 < 0.26.
Adding and Subtracting Decimals
- When adding and subtracting decimals, you use the same way that you add and subtract whole numbers. The decimal points and the place values must be lined up correctly. You must line up the tenths with the tenths, the hundredths with the hundredths, and the thousandths with the thousandths. Add each column starting from the most right side. For example,
  1.12 + 0.23 = ?
  1 . 1 2
  + 0 . 2 3
  1 . 3 5
  
  0.43 + 0.455 = ?
  0 . 4 3
  + 0 . 4 5 5
  0 . 8 8 5
Rounding Decimals: the same way with the rounding whole numbers
Round a decimal to the nearest tenths:
- If the digit of the hundredths place of a decimal is four or less, round down the decimal to the tenth place and the digit of the tenths place does not change. For example, rounding 0.843 to the nearest tenths would give 0.8.
- If the digit of the hundredths place of a decimal is five to nine, round up the decimal to the tenth place and the digit of the tenth place is increased by 1. For example, rounding 0.953 to the nearest tenths would give 1.0.
Round a decimal to the nearest hundredths:
- If the digit of the thousandths place of the decimal is four or less, round the decimal down to the hundredths place and the digit of the hundredths place is not changed. For example, rounding 0.843 to the nearest hundredths would give 0.84.
- If the digit of the thousandths place of a decimal is five to nine, round up the decimal to the hundredths place and the digit of hundredths place is increased by one. For example, rounding 0.658 to the nearest hundredths would give 0.66.
General rule to round a decimal:
- Retain the asked number of decimal places (e.g. 1 for tenths, 2 for hundredths, 3 for thousandths, ...)
- If the next decimal place value is 5 or more, increase the value in the last retained decimal place by 1.

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