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Numbers - Fraction

SOURCE:COMPETITION
Number of Problems: 9.
FOR PRINT ::: (Book)

Problem Num : 1

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Let $n$ be a positive integer such that $frac 12 + frac 13 + frac 17 + frac 1n$ is an integer. Which of the following statements is not true:
$mathrm{(A)} 2 ext{divides }nqquadmathrm{(B)} 3 ext{divides }nqquadmathrm{(C)}$ $6 ext{divides }n qquadmathrm{(D)} 7 ext{divides }nqquadmathrm{(E)} n > 84$
'

Category Fraction

Analysis

Solution/Hint
Since $frac 12 + frac 13 + frac 17 = frac {41}{42}$ ,
$0 < lim_{n ightarrow infty} left(frac{41}{42} + frac{1}{n} ight) < frac {41}{42} + frac 1n < frac{41}{42}...$
From which it follows that $frac{41}{42} + frac 1n = 1$ and $n = 42$ . Thus the answer is $oxed{mathrm{(E)} n>84}$ .

Problem Num : 2

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Which of the following is the same as $dfrac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}$ ?
$extbf{(A)} -1 qquad extbf{(B)} -frac23 qquad extbf{(C)} frac23 qquad extbf{(D)} 1 qquad extbf{(E)} frac{14}...$
'

Category Fraction

Analysis

Solution/Hint
We can rewrite the given fraction as:
$frac{-3(2)+2(7)}{3(-3)+3(7)}=frac{2(4)}{3(4)}=frac{2}{3} longrightarrow oxed{ extbf{(C)}}$

Problem Num : 3

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Given that $-4leq xleq-2$ and $2leq yleq4$ , what is the largest possible value of $frac{x+y}{x}$ ?
$mathrm{(A) } -1 qquad mathrm{(B) } -frac12 qquad mathrm{(C) } 0 qquad mathrm{(D) } frac12 qquad mathrm{(E)...$
'

Category Fraction

Analysis

Solution/Hint
Rewrite $frac{(x+y)}x$ as $frac{x}x+frac{y}x=1+frac{y}x$ .
We also know that $frac{y}x<0$ because $x$ and $y$ are of opposite sign.
Therefore, $1+frac{y}x$ is maximized when $frac{y}x$ is minimized, which occurs when $|x|$ is the largest and $|y|$ is the smallest.
This occurs at (-4,2), so $frac{x+y}x=1-frac12=frac12Rightarrow mathrm{(D)}$ .

Problem Num : 4

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Which of the following is equal to $dfrac{frac{1}{3}-frac{1}{4}}{frac{1}{2}-frac{1}{3}}$ ?
$ext{(A) } frac 14qquad ext{(B) } frac 13qquad ext{(C) } frac 12qquad ext{(D) } frac 23qquad ext{(E) } frac 34$
'

Category Fraction

Analysis

Solution/Hint
Multiplying the numerator and the denominator by the same value does not change the value of the fraction. We can multiply both by $12$ , getting $dfrac{4-3}{6-4} = oxed{dfrac 12}$ .
Alternately, we can directly compute that the numerator is $dfrac 1{12}$ , the denominator is $dfrac 16$ , and hence their ratio is $oxed{frac {1} {2}}$ .

Problem Num : 5

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Which of the following is equal to the product $frac{8}{4}cdotfrac{12}{8}cdotfrac{16}{12}cdotcdotscdotfrac{4n+4}{4n}cdotcdotscdotfrac{2008}{2004}?$
$extbf{(A)} 251qquad extbf{(B)} 502qquad extbf{(C)} 1004qquad extbf{(D)} 2008qquad extbf{(E)} 4016$

Contents

1 Problem

2 Solution

2.1 Solution 1

2.2 Solution 2

3 See Also

'

Category Fraction

Analysis

Solution/Hint
Solution 1

$frac {8}{4}cdotfrac {12}{8}cdotfrac {16}{12}cdotsfrac {4n + 4}{4n}cdotsfrac {2008}{2004} = frac {1}{4}cdotleft(f...$ $502 Rightarrow B$ .

Solution 2

Notice that everything cancels out except for $2008$ in the numerator and $4$ in the denominator.
Thus, the product is $frac{2008}{4}=502$ , and the answer is $extbf{(B)}$ .

Problem Num : 6

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
Which of the following is equal to $1 + frac {1}{1 + frac {1}{1 + 1}}$ ?
$extbf{(A)} frac {5}{4} qquad extbf{(B)} frac {3}{2} qquad extbf{(C)} frac {5}{3} qquad extbf{(D)} 2 qquad ...$
'

Category Fraction

Analysis

Solution/Hint
We compute:
$egin{align*}1 + frac {1}{1 + frac {1}{1 + 1}}&=1 + frac {1}{1 + frac {1}{1 + 1}}\&=1 + frac {1}{1 + frac 12}...$
This is choice $oxed{ ext{C}}$ .

Problem Num : 7

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
What is $dfrac{2+4+6}{1+3+5} - dfrac{1+3+5}{2+4+6} ?$
$extbf{(A)} -1qquad extbf{(B)} frac{5}{36}qquad extbf{(C)} frac{7}{12}qquad extbf{(D)} frac{147}{60}qquad ext...$
'

Category Fraction

Analysis

Solution/Hint
First, simplify the fractions.
$dfrac{2+4+6}{1+3+5} - dfrac{1+3+5}{2+4+6} = dfrac{12}{9} - dfrac{9}{12}$
$dfrac{12}{9} - dfrac{9}{12} = dfrac{48}{36} - dfrac{27}{36} = dfrac{21}{36} = oxed{dfrac{7}{12} extbf{(C)}}$

Problem Num : 8

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
On Halloween Casper ate $frac{1}{3}$ of his candies and then gave $2$ candies to his brother. The next day he ate $frac{1}{3}$ of his remaining candies and then gave $4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning?
$extbf{(A)} 30 qquad extbf{(B)} 39 qquad extbf{(C)} 48 qquad extbf{(D)} 57 qquad extbf{(E)} 66$
'

Category Fraction

Analysis

Solution/Hint
First, simplify the fractions.
$dfrac{2+4+6}{1+3+5} - dfrac{1+3+5}{2+4+6} = dfrac{12}{9} - dfrac{9}{12}$
$dfrac{12}{9} - dfrac{9}{12} = dfrac{48}{36} - dfrac{27}{36} = dfrac{21}{36} = oxed{dfrac{7}{12} extbf{(C)}}$

Problem Num : 9

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
The number halfway between $dfrac{1}{6}$ and $dfrac{1}{4}$ is
$ext{(A)} dfrac{1}{10} qquad ext{(B)} dfrac{1}{5} qquad ext{(C)} dfrac{5}{24} qquad ext{(D)} dfrac{7}{24} ...$
'

Category Fraction

Analysis

Solution/Hint
The number halfway between is the average.
$frac{frac16 + frac14}{2} = frac{frac{2}{12} + frac{3}{12}}{2} = oxed{ ext{(C)} frac{5}{24}}$