Set F = 9, 12, 15, 20, 27 and E = multiples of 3 between 5 and 20. Write down n(F E). Make m the subject of y…

Question

Set F = 9, 12, 15, 20, 27 and E = multiples of 3 between 5 and 20. Write down n(F E). Make m the subject of y = (cm - 2b)/t. Simplify (2a+b)/3 - (ab+a+b)/4.

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: Set F and Set E
Given that Set F = $\{9, 12, 15, 20, 27\}$ and E = {multiples of 3 between 5 and 20}. Write down $n(F \cap E)$ .

List the elements of Set E.
The multiples of 3 between 5 and 20 are 6, 9, 12, 15, 18.
So, $E = \{6, 9, 12, 15, 18\}$ .
Find the intersection of Set F and Set E ( $F \cap E$ ).
The elements common to both sets F and E are 9, 12, and 15.
So, $F \cap E = \{9, 12, 15\}$ .
Determine the number of elements in $F \cap E$ .
The number of elements in $F \cap E$ is 3.
$n(F \cap E) = 3$
The number of elements is $\boxed{\text{3}}$ .

: Make m the subject of the formula
Given the formula $y = \frac{cm - 2b}{t}$ .
Multiply both sides of the equation by $t$ .
$yt = cm - 2b$
Add $2b$ to both sides of the equation.
$yt + 2b = cm$
Divide both sides by $c$ .
$\frac{yt + 2b}{c} = m$
So, $m = \boxed{\frac{yt + 2b}{c}}$ .

: Simplify $\frac{2a+b}{3} - \frac{ab+a+b}{4}$
Find a common denominator for the two fractions, which is 12.
$\frac{4(2a+b)}{12} - \frac{3(ab+a+b)}{12}$
Expand the numerators.
$\frac{8a+4b}{12} - \frac{3ab+3a+3b}{12}$
Combine the fractions over the common denominator.
$\frac{8a+4b - (3ab+3a+3b)}{12}$ $\frac{8a+4b - 3ab - 3a - 3b}{12}$
Combine like terms in the numerator.
$\frac{(8a-3a) + (4b-3b) - 3ab}{12}$ $\frac{5a + b - 3ab}{12}$
The simplified expression is $\boxed{\frac{5a + b - 3ab}{12}}$ .

: Simplify $(\frac{2}{3x})^{-1} \times \frac{6}{x}$
Apply the negative exponent rule $(A/B)^{-1} = B/A$ .
$\left(\frac{2}{3x}\right)^{-1} = \frac{3x}{2}$
Multiply the result by $\frac{6}{x}$ .
$\frac{3x}{2} \times \frac{6}{x}$
Cancel out common terms ( $x$ in the numerator and denominator).
$\frac{3 \times 6}{2}$
Perform the multiplication and division.
$\frac{18}{2} = 9$
The simplified expression is $\boxed{\text{9}}$ .

: Find the first three terms of the sequence with $n^{th}$ term $3n^2-1$ .
Find the first term by substituting $n=1$ .
$3(1)^2 - 1 = 3(1) - 1 = 3 - 1 = 2$
Find the second term by substituting $n=2$ .
$3(2)^2 - 1 = 3(4) - 1 = 12 - 1 = 11$
Find the third term by substituting $n=3$ .
$3(3)^2 - 1 = 3(9) - 1 = 27 - 1 = 26$
The first three terms are $\boxed{\text{2, 11, 26}}$ .

: Solve the equation $r - 2\frac{1}{2} = -\frac{1}{8}$
Convert the mixed number to an improper fraction.
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$
The equation becomes $r - \frac{5}{2} = -\frac{1}{8}$ .
Add $\frac{5}{2}$ to both sides of the equation.
$r = -\frac{1}{8} + \frac{5}{2}$
Find a common denominator for the fractions, which is 8.
$r = -\frac{1}{8} + \frac{5 \times 4}{2 \times 4}$ $r = -\frac{1}{8} + \frac{20}{8}$
Add the fractions.
$r = \frac{-1 + 20}{8}$ $r = \frac{19}{8}$
The solution is $r = \boxed{\frac{19}{8}}$ or $r = \boxed{2\frac{3}{8}}$ .

Set F = 9, 12, 15, 20, 27 and E = multiples of 3 between 5 and 20. Write down n(F E). Make m the subject of y = (cm − 2b)/t. Simplify (2a+b)/3 - (ab+a+b)/4.

Step-by-step solution

List the elements of Set E.

Find the intersection of Set F and Set E ( $F \cap E$ ).

Determine the number of elements in $F \cap E$ .

Multiply both sides of the equation by $t$ .

Add $2b$ to both sides of the equation.

Divide both sides by $c$ .

Find a common denominator for the two fractions, which is 12.

Expand the numerators.

Combine the fractions over the common denominator.

Combine like terms in the numerator.

Apply the negative exponent rule $(A/B)^{-1} = B/A$ .

Multiply the result by $\frac{6}{x}$ .

Cancel out common terms ( $x$ in the numerator and denominator).

Perform the multiplication and division.

Find the first term by substituting $n=1$ .

Find the second term by substituting $n=2$ .

Find the third term by substituting $n=3$ .

Convert the mixed number to an improper fraction.

Add $\frac{5}{2}$ to both sides of the equation.

Find a common denominator for the fractions, which is 8.

Add the fractions.

Still stuck on this one?

More Mathematics Questions

Set F = 9, 12, 15, 20, 27 and E = multiples of 3 between 5 and 20. Write down n(F E). Make m the subject of y = (cm − 2b)/t. Simplify (2a+b)/3 - (ab+a+b)/4.

Step-by-step solution

List the elements of Set E.

Find the intersection of Set F and Set E (F∩EF \cap EF∩E).

Determine the number of elements in F∩EF \cap EF∩E.

Multiply both sides of the equation by ttt.