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Equation - System Of Equation

For COMPETITION
Number of Total Problems: 10.
FOR PRINT ::: (Book)

Problem Num : 1

From : NCTM

Type: None

Section:Equation

Theme:Num of Equations
Adjustment# : 0

Difficulty: 1

Category System Of Equation

Analysis

Solution/Answer

Problem Num : 2

From : NCTM

Type: None

Section:Equation

Theme:None
Adjustment# : 0

Difficulty: 1

Category System Of Equation

Analysis

Solution/Answer

Problem Num : 3

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

'
Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$ . What is $a + b + c + d$ ?
$ext{(A)} -5 qquad ext{(B)} -10/3 qquad ext{(C)} -7/3 qquad ext{(D)} 5/3 qquad ext{(E)} 5$
'

Category System Of Equation

Analysis

Solution/Answer
Let $x=a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$ . Since one of the sums involves a, b, c, and d, it makes sense to consider 4x. We have $4x=(a+1)+(b+2)+(c+3)+(d+4)=a+b+c+d+10=4(a+b+c+d)+20$ . Rearranging, we have $3(a+b+c+d)=-10$ , so $a+b+c+d=frac{-10}{3}$ . Thus, our answer is $oxed{ ext{(B)} -10/3}$ .

Answer:

Problem Num : 4

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

The sum of three numbers is $20$ . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
$mathrm{(A) } 28qquad mathrm{(B) } 40qquad mathrm{(C) } 100qquad mathrm{(D) } 400qquad mathrm{(E) } 800$
'>''
The sum of three numbers is $20$ . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
$mathrm{(A) } 28qquad mathrm{(B) } 40qquad mathrm{(C) } 100qquad mathrm{(D) } 400qquad mathrm{(E) } 800$
'

Category System Of Equation

Analysis

Solution/Answer
Solution 1

Let the numbers be $x$ , $y$ , and $z$ in that order. The given tells us that
$egin{eqnarray*}y&=&7z\x&=&4(y+z)=4(7z+z)=4(8z)=32z\x+y+z&=&32z+7z+z=40z=20\z&=&frac{20}...$
Therefore, the product of all three numbers is $xyz=16cdotfrac{7}{2}cdotfrac{1}{2}=28 Rightarrow mathrm{(A)}$ .

Solution 2

Alternatively, we can set up the system in matrix form:
$egin{eqnarray*}1x+1y+1z&=&20\1x-4y-4z&=&0\0x+1y-7z&=&0\end{eqnarray*}$
Or, in matrix form $egin{bmatrix}1 & 1 & 1 \1 & -4 & -4 \0 & 1 & -7end{bmatrix}egin{bmatrix}x \y \z \end{bmatr...$
To solve this matrix equation, we can rearrange it thus:
$egin{bmatrix}x \y \z \end{bmatrix}= egin{bmatrix}1 & 1 & 1 \1 & -4 & -4 \0 & 1 & -7end{bma...$
Solving this matrix equation by using inverse matrices and matrix multiplication yields
$egin{bmatrix}x \y \z \end{bmatrix} =egin{bmatrix}frac{1}{2} \frac{7}{2} \16 \end{bmatrix}$
Which means that $x = frac{1}{2}$ , $y = frac{7}{2}$ , and $z = 16$ . Therefore, $xyz = frac{1}{2}cdotfrac{7}{2}cdot16 = 28$

Answer:

Problem Num : 5

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

'
A school store sells 7 pencils and 8 notebooks for $mathdollar 4.15$ . It also sells 5 pencils and 3 notebooks for $mathdollar 1.77$ . How much do 16 pencils and 10 notebooks cost?
$ext{(A)}mathdollar 1.76 qquad ext{(B)}mathdollar 5.84 qquad ext{(C)}mathdollar 6.00 qquad ext{(D)}mathdollar 6...$
'

Category System Of Equation

Analysis

Solution/Answer
We let $p =$ cost of pencils in cents, $n =$ number of notebooks in cents. Then
$egin{align*}7p + 8n = 415 &Longrightarrow 35p + 40n = 2075\5p + 3n = 177 &Longrightarrow 35p + 21n = 1239end{...$
Subtracting these equations yields $19n = 836 Longrightarrow n = 44$ . Backwards solving gives $p = 9$ . Thus the answer is $16p + 10n = 584 mathrm{(B)}$ .

Answer:

Problem Num : 6

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

'
Suppose that $frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $frac{1}{2}$ of $5$ bananas?
$mathrm{(A)} 2qquadmathrm{(B)} frac{5}{2}qquadmathrm{(C)} 3qquadmathrm{(D)} frac{7}{2}qquadmathrm{(E)} 4$
'

Category System Of Equation

Analysis

Solution/Answer
If $frac{2}{3}cdot10 ext{bananas}=8 ext{oranges}$ , then $frac{1}{2}cdot5 ext{bananas}=left(frac{1}{2}cdot 5 ext{bananas} ight)cdotleft(frac{8 ext{oranges}}{frac{2}...$ .

Answer:

Problem Num : 7

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

'
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
$extbf{(A)} frac{10}{3}qquad extbf{(B)} frac{20}{3}qquad extbf{(C)} 7qquad extbf{(D)} frac{15}{2}qquad extbf{...$
'

Category System Of Equation

Analysis

Solution/Answer
Let the two numbers equal $x$ and $y$ . From the information given in the problem, two equations can be written:
$xy=9$
$frac{1}{x}=4 left( frac{1}{y} ight)$
Therefore, $4x=y$
Replacing $y$ with $4x$ in the equation,
$4x^2=9$
So $x=frac{3}{2}$ and $y$ would then be $4 imes$ $frac{3}{2}=6$
The sum would be $frac{3}{2}+6$ = $oxed{ extbf{(D)} frac{15}{2}}$

Answer:

Problem Num : 8

From : NCTM

Type: Calculation

Section:Equation

Theme:None
Adjustment# : 0

Difficulty: 1

Category System Of Equation

Analysis

Solution/Answer ''
Answer:

Problem Num : 9

From : AMC10

Type:

Section:Equation

Theme:
Adjustment# : 0

Difficulty: 1

'
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
$extbf{(A)} 4qquad extbf{(B)} 5qquad extbf{(C)} 6qquad extbf{(D)} 7qquad extbf{(E)} 8$
'

Category System Of Equation

Analysis

Solution/Answer
Let the three numbers be equal to $a$ , $b$ , and $c$ . We can now write three equations:
$a+b=12$
$b+c=17$
$a+c=19$
Adding these equations together, we get that
$2(a+b+c)=48$ and
$a+b+c=24$
Substituting the original equations into this one, we find
$c+12=24$
$a+17=24$
$b+19=24$
Therefore, our numbers are 12, 7, and 5. The middle number is $oxed{ extbf{(D)} 7}$

Answer:

Problem Num : 10

From : NCTM

Type: Calculation

Section:Equation

Theme:None
Adjustment# : 0

Difficulty: 1

Category System Of Equation

Analysis

Solution/Answer

Array ( [0] => 5945 [1] => 5953 [2] => 7761 [3] => 7779 [4] => 7847 [5] => 7865 [6] => 7926 [7] => 3063 [8] => 7928 [9] => 3065 ) 10