Evaluate the expression: (9 1/2 − 3 1/3 ÷ 5/9) / (3/5 of 6 1/4 + 1 1/2)

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Here are the solutions to the problems.

:

Convert all mixed numbers to improper fractions.
$9\frac{1}{2} = \frac{19}{2}$ $3\frac{1}{3} = \frac{10}{3}$ $6\frac{1}{4} = \frac{25}{4}$ $1\frac{1}{2} = \frac{3}{2}$
Evaluate the numerator.
The numerator is $9\frac{1}{2} - 3\frac{1}{3} \div \frac{5}{9}$ .
Substitute the improper fractions:
$\frac{19}{2} - \frac{10}{3} \div \frac{5}{9}$
Perform the division first:
$\frac{10}{3} \div \frac{5}{9} = \frac{10}{3} \times \frac{9}{5} = \frac{10 \times 9}{3 \times 5} = \frac{90}{15} = 6$
Now, perform the subtraction:
$\frac{19}{2} - 6 = \frac{19}{2} - \frac{12}{2} = \frac{19 - 12}{2} = \frac{7}{2}$
Evaluate the denominator.
The denominator is $\frac{3}{5} \text{ of } 6\frac{1}{4} + 1\frac{1}{2}$ .
"of" means multiplication. Substitute the improper fractions:
$\frac{3}{5} \times \frac{25}{4} + \frac{3}{2}$
Perform the multiplication first:
$\frac{3}{5} \times \frac{25}{4} = \frac{3 \times 25}{5 \times 4} = \frac{75}{20} = \frac{15}{4}$
Now, perform the addition:
$\frac{15}{4} + \frac{3}{2} = \frac{15}{4} + \frac{6}{4} = \frac{15 + 6}{4} = \frac{21}{4}$
Divide the numerator by the denominator.
$\frac{\frac{7}{2}}{\frac{21}{4}} = \frac{7}{2} \times \frac{4}{21} = \frac{7 \times 4}{2 \times 21} = \frac{28}{42}$
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 14:
$\frac{28 \div 14}{42 \div 14} = \frac{2}{3}$
The evaluated expression is $\boxed{\frac{2}{3}}$ .

:
Express all terms in the equation with a base of 2.
The equation is $32^{(x-3)} \times 8^{(x+4)} = 64 \div 2^x$ .
$32 = 2^5$ $8 = 2^3$ $64 = 2^6$
Substitute these into the equation:
$(2^5)^{(x-3)} \times (2^3)^{(x+4)} = 2^6 \div 2^x$
Apply the exponent rules $(a^m)^n = a^{mn}$ , $a^m \times a^n = a^{m+n}$ , and $a^m \div a^n = a^{m-n}$ .
$2^{5(x-3)} \times 2^{3(x+4)} = 2^{6-x}$ $2^{5x-15} \times 2^{3x+12} = 2^{6-x}$ $2^{(5x-15) + (3x+12)} = 2^{6-x}$ $2^{8x-3} = 2^{6-x}$
Equate the exponents since the bases are the same.
$8x - 3 = 6 - x$
Solve for $x$ .
Add $x$ to both sides:
$8x + x - 3 = 6$ $9x - 3 = 6$
Add 3 to both sides:
$9x = 6 + 3$ $9x = 9$
Divide by 9:
$x = \frac{9}{9}$ $x = 1$
The value of $x$ is $\boxed{1}$ .

:
Set the recurring decimal equal to $x$ .
Let $x = 17.80\dot{4}$ , which means $x = 17.80444...$
Multiply $x$ by a power of 10 to move the non-repeating part to the left of the decimal point.
The non-repeating part is '80'. There are two digits.
$100x = 1780.444... \quad (\text{Equation 1})$
Multiply $x$ by a power of 10 to move one full repeating block to the left of the decimal point, along with the non-repeating part.
The repeating part is '4'. There is one repeating digit. So, we multiply by $10^3 = 1000$ .
$1000x = 17804.444... \quad (\text{Equation 2})$
Subtract Equation 1 from Equation 2.
$1000x - 100x = 17804.444... - 1780.444...$ $900x = 16024$
Solve for $x$ and simplify the fraction.
$x = \frac{16024}{900}$
Divide both the numerator and the denominator by their greatest common divisor. Both are divisible by 4.
$x = \frac{16024 \div 4}{900 \div 4} = \frac{4006}{225}$
Convert the improper fraction to a mixed fraction.
Divide 4006 by 225:
$4006 \div 225 = 17 \text{ with a remainder}$ $17 \times 225 = 3825$ $\text{Remainder} = 4006 - 3825 = 181$
So, the mixed fraction is $17\frac{181}{225}$ .
The fraction $\frac{181}{225}$ is in its simplest form because 181 is a prime number and 225 is not divisible by 181.
The recurring decimal as a mixed fraction is $\boxed{17\frac{181}{225}}$ .

Evaluate the expression: (9 1/2 - 3 1/3 ÷ 5/9) / (3/5 of 6 1/4 + 1 1/2)

Step-by-step solution

Convert all mixed numbers to improper fractions.

Evaluate the numerator.

Evaluate the denominator.

Divide the numerator by the denominator.

Express all terms in the equation with a base of 2.

Apply the exponent rules $(a^m)^n = a^{mn}$ , $a^m \times a^n = a^{m+n}$ , and $a^m \div a^n = a^{m-n}$ .

Equate the exponents since the bases are the same.

Solve for $x$ .

Set the recurring decimal equal to $x$ .

Multiply $x$ by a power of 10 to move the non-repeating part to the left of the decimal point.

Multiply $x$ by a power of 10 to move one full repeating block to the left of the decimal point, along with the non-repeating part.

Subtract Equation 1 from Equation 2.

Solve for $x$ and simplify the fraction.

Convert the improper fraction to a mixed fraction.

Still stuck on this one?

More Mathematics Questions

Evaluate the expression: (9 1/2 - 3 1/3 ÷ 5/9) / (3/5 of 6 1/4 + 1 1/2)

Step-by-step solution

Convert all mixed numbers to improper fractions.

Evaluate the numerator.

Evaluate the denominator.

Divide the numerator by the denominator.

Express all terms in the equation with a base of 2.

Apply the exponent rules (am)n=amn(a^m)^n = a^{mn}(am)n=amn, am×an=am+na^m \times a^n = a^{m+n}am×an=am+n, and am÷an=am−na^m \div a^n = a^{m-n}am÷an=am−n.

Equate the exponents since the bases are the same.

Solve for xxx.

Set the recurring decimal equal to xxx.

Multiply xxx by a power of 10 to move the non-repeating part to the left of the decimal point.

Multiply xxx by a power of 10 to move one full repeating block to the left of the decimal point, along with the non-repeating part.

Subtract Equation 1 from Equation 2.

Solve for xxx and simplify the fraction.

Convert the improper fraction to a mixed fraction.

Still stuck on this one?

More Mathematics Questions

Apply the exponent rules $(a^m)^n = a^{mn}$ , $a^m \times a^n = a^{m+n}$ , and $a^m \div a^n = a^{m-n}$ .

Solve for $x$ .

Set the recurring decimal equal to $x$ .

Multiply $x$ by a power of 10 to move the non-repeating part to the left of the decimal point.

Multiply $x$ by a power of 10 to move one full repeating block to the left of the decimal point, along with the non-repeating part.

Solve for $x$ and simplify the fraction.