PART I: OBJECTIVE (50 POINTS) Answer all the questions. Each question is followed by four options lettered A…

Question

PART I: OBJECTIVE (50 POINTS) Answer all the questions. Each question is followed by four options lettered A to D. Circle the letter bearing the best option or answer.

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Here are the solutions to the math problems:

1) The numeral is 69,425,950.
The digit 4 is in the hundreds of thousands place.
The correct option is C.

2) To divide 1.68 by 1.2:

Write the division as a fraction.
$\frac{1.68}{1.2}$
Multiply the numerator and denominator by 10 to remove the decimal from the denominator.
$\frac{1.68 \times 10}{1.2 \times 10} = \frac{16.8}{12}$
Perform the division.
$16.8 \div 12 = 1.4$
The correct option is A.

3) To find the LCM of 27, 90, and 84:
Find the prime factorization of each number.
$27 = 3^3$ $90 = 2 \times 3^2 \times 5$ $84 = 2^2 \times 3 \times 7$
Take the highest power of each prime factor present in any of the numbers.
$\text{LCM} = 2^2 \times 3^3 \times 5^1 \times 7^1$
Calculate the product.
$\text{LCM} = 4 \times 27 \times 5 \times 7 = 108 \times 35 = 3780$
The correct option is D.

4) Let the certain number be $N$ .
Set up the equation based on the given information.
$2\% \text{ of } N = \frac{3}{4}$
Convert the percentage to a decimal and the fraction to a decimal.
$0.02 \times N = 0.75$
Solve for $N$ .
$N = \frac{0.75}{0.02}$ $N = \frac{75}{2}$ $N = 37.5$
The correct option is C.

5) For a polygon with $n$ vertices, the number of diagonals that can be drawn from a single vertex is given by the formula $n-3$ .
Identify the number of vertices for a pentagon.
A pentagon has $n=5$ vertices.
Apply the formula.
$\text{Number of diagonals} = 5 - 3 = 2$
The correct option is B.

6) To simplify $8 - 6(2x + 3(4-x) - 1)$ :
Simplify the innermost parenthesis.
$8 - 6(2x + 12 - 3x - 1)$
Combine like terms inside the main parenthesis.
$8 - 6(-x + 11)$
Distribute the $-6$ .
$8 + 6x - 66$
Combine the constant terms.
$6x - 58$
The correct option is A.

7) To solve for $x$ in $3(x+4) + 5(2x-1) = 4(2x+8) + 5$ :
Distribute the numbers into the parentheses.
$3x + 12 + 10x - 5 = 8x + 32 + 5$
Combine like terms on each side of the equation.
$13x + 7 = 8x + 37$
Move terms with $x$ to one side and constants to the other.
$13x - 8x = 37 - 7$ $5x = 30$
Divide to solve for $x$ .
$x = \frac{30}{5}$ $x = 6$
The correct option is B.

8) To make $x$ the subject of $ax - b = bx$ :
Move all terms containing $x$ to one side of the equation.
$ax - bx = b$
Factor out $x$ .
$x(a - b) = b$
Divide by $(a-b)$ to isolate $x$ .
$x = \frac{b}{a-b}$
The correct option is D.

9) To factorize $U^2 - 16V^2$ :
Recognize this as a difference of squares, $A^2 - B^2 = (A-B)(A+B)$ .
Here, $A = U$ and $B = \sqrt{16V^2} = 4V$ .
Apply the difference of squares formula.
$U^2 - (4V)^2 = (U - 4V)(U + 4V)$
The correct option is A.

10) To factorize $3x^2+5x-2$ :
Look for two binomials $(Ax+B)(Cx+D)$ that multiply to the given quadratic.
Since the coefficient of $x^2$ is 3, the factors must be of the form $(3x+B)(x+D)$ .
Find factors of $-2$ (B and D) such that the sum of the inner and outer products equals $5x$ .
Let's try $(3x-1)(x+2)$ :
$(3x-1)(x+2) = 3x(x) + 3x(2) - 1(x) - 1(2)$ $= 3x^2 + 6x - x - 2$ $= 3x^2 + 5x - 2$
This matches the original expression.
The correct option is A.

11) To express the ratio $1 \frac{1}{2} : 2 \frac{1}{2}$ in the simplest form:
Convert the mixed numbers to improper fractions.
$1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$ $2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$
Write the ratio with improper fractions.
$\frac{3}{2} : \frac{5}{2}$
Multiply both parts of the ratio by the common denominator, 2, to eliminate the fractions.
$\left(\frac{3}{2}\right) \times 2 : \left(\frac{5}{2}\right) \times 2$ $3 : 5$
The correct option is B.

12) If $y$ varies directly as $x$ , the relationship is $y = kx$ , where $k$ is the constant of proportionality.
Use the given values ( $y=100$ when $x=20$ ) to find $k$ .
$100 = k \times 20$ $k = \frac{100}{20}$ $k = 5$
Use the value of $k$ to find $y$ when $x=25$ .
$y = 5 \times 25$ $y = 125$
The correct option is D.

PART I: OBJECTIVE (50 POINTS) Answer all the questions. Each question is followed by four options lettered A to D. Circle the letter bearing the best option or answer.

Answer

Step-by-step solution

Write the division as a fraction.

Multiply the numerator and denominator by 10 to remove the decimal from the denominator.

Perform the division.

Find the prime factorization of each number.

Take the highest power of each prime factor present in any of the numbers.

Calculate the product.

Set up the equation based on the given information.

Convert the percentage to a decimal and the fraction to a decimal.

Solve for $N$ .

Identify the number of vertices for a pentagon.

Apply the formula.

Simplify the innermost parenthesis.

Combine like terms inside the main parenthesis.

Distribute the $-6$ .

Combine the constant terms.

Distribute the numbers into the parentheses.

Combine like terms on each side of the equation.

Move terms with $x$ to one side and constants to the other.

Divide to solve for $x$ .

Move all terms containing $x$ to one side of the equation.

Factor out $x$ .

Divide by $(a-b)$ to isolate $x$ .

Recognize this as a difference of squares, $A^2 - B^2 = (A-B)(A+B)$ .

Apply the difference of squares formula.

Look for two binomials $(Ax+B)(Cx+D)$ that multiply to the given quadratic.

Find factors of $-2$ (B and D) such that the sum of the inner and outer products equals $5x$ .

Convert the mixed numbers to improper fractions.

Write the ratio with improper fractions.

Multiply both parts of the ratio by the common denominator, 2, to eliminate the fractions.

Use the given values ( $y=100$ when $x=20$ ) to find $k$ .

Use the value of $k$ to find $y$ when $x=25$ .

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PART I: OBJECTIVE (50 POINTS) Answer all the questions. Each question is followed by four options lettered A to D. Circle the letter bearing the best option or answer.

Answer

Step-by-step solution

Write the division as a fraction.

Multiply the numerator and denominator by 10 to remove the decimal from the denominator.

Perform the division.

Find the prime factorization of each number.

Take the highest power of each prime factor present in any of the numbers.

Calculate the product.

Set up the equation based on the given information.

Convert the percentage to a decimal and the fraction to a decimal.

Solve for NNN.

Identify the number of vertices for a pentagon.

Apply the formula.

Simplify the innermost parenthesis.

Combine like terms inside the main parenthesis.

Distribute the −6-6−6.

Combine the constant terms.

Distribute the numbers into the parentheses.

Combine like terms on each side of the equation.

Move terms with xxx to one side and constants to the other.

Divide to solve for xxx.

Move all terms containing xxx to one side of the equation.

Factor out xxx.

Divide by (a−b)(a-b)(a−b) to isolate xxx.

Recognize this as a difference of squares, A2−B2=(A−B)(A+B)A^2 - B^2 = (A-B)(A+B)A2−B2=(A−B)(A+B).

Apply the difference of squares formula.

Look for two binomials (Ax+B)(Cx+D)(Ax+B)(Cx+D)(Ax+B)(Cx+D) that multiply to the given quadratic.

Find factors of −2-2−2 (B and D) such that the sum of the inner and outer products equals 5x5x5x.

Convert the mixed numbers to improper fractions.

Write the ratio with improper fractions.

Multiply both parts of the ratio by the common denominator, 2, to eliminate the fractions.

Use the given values (y=100y=100y=100 when x=20x=20x=20) to find kkk.

Use the value of kkk to find yyy when x=25x=25x=25.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Solve for $N$ .

Distribute the $-6$ .

Move terms with $x$ to one side and constants to the other.

Divide to solve for $x$ .

Move all terms containing $x$ to one side of the equation.

Factor out $x$ .

Divide by $(a-b)$ to isolate $x$ .

Recognize this as a difference of squares, $A^2 - B^2 = (A-B)(A+B)$ .

Look for two binomials $(Ax+B)(Cx+D)$ that multiply to the given quadratic.

Find factors of $-2$ (B and D) such that the sum of the inner and outer products equals $5x$ .

Use the given values ( $y=100$ when $x=20$ ) to find $k$ .

Use the value of $k$ to find $y$ when $x=25$ .