PSAT Math - How to divide fractions

Define an operation $\otimes ;$ as follows:

For all real numbers ,

$a \otimes b = \frac{4-a}{b}$ .

Evaluate $1 \frac{1}{2} \otimes 2 \frac{1}{3}$ .

$1 \frac{1}{14}$

$5\frac{5}{6}$

$1\frac{1}{6}$

$\frac{14}{15}$

$\frac{6}{35}$

Explanation

$a \otimes b = \frac{4-a}{b}$ , or, equivalently,

$a \otimes b =( 4-a ) \div b$

$1 \frac{1}{2} \otimes 2 \frac{1}{3} =\left ( 4-1 \frac{1}{2} \right ) \div 2 \frac{1}{3}$

$= 2 \frac{1}{2} \div 2 \frac{1}{3}$

$= \frac{5}{2} \div \frac{7}{3}$

$= \frac{5}{2} \times \frac{3}{7}$

$= \frac{15}{14}$

$= 1 \frac{1}{14}$

Define an operation $\ddagger$ as follows:

For all real numbers ,

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ .

Evaluate $\frac{3}{4} \ddagger\frac{4}{3}$ .

$-1 \frac{31}{144}$

$1 \frac{31}{144}$

$3\frac{1}{12}$

$-3\frac{1}{12}$

Explanation

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ ,

or, equivalently,

$a \ddagger b = (a \div b) - (b \div a)$

$\frac{3}{4} \ddagger\frac{4}{3} =\left (\frac{3}{4} \div \frac{4}{3} \right ) - \left (\frac{4}{3} \div \frac{3}{4} \right )$

$=\left (\frac{3}{4} \times \frac{3}{4} \right ) - \left (\frac{4}{3} \times \frac{4} {3}\right )$

$= \frac{9}{16} -\frac{16}{9}$

From here we need to find a common denominator.

$=\frac{9}{16}\bigg(\frac{9}{9}\bigg)- \frac{16}{9}\bigg(\frac{16}{16}\bigg)$

$= \frac{81}{144} -\frac{256}{144}$

$= -\frac{175}{144}$

$= -1 \frac{31}{144}$

Define an operation $\odot$ as follows:

For all real numbers ,

$a \odot b = \frac{a}{b+2}$ .

Evaluate $\frac{3}{5} \odot \frac{1}{5}$ .

$\frac{3} {11}$

$1\frac{8}{25}$

$3\frac{2}{3}$

$\frac{25}{33}$

Explanation

$a \odot b = \frac{a}{b+2}$ ,

or, equivalently,

$a \odot b = a \div\left ( b+2 \right )$

$\frac{3}{5} \odot \frac{1}{5}= \frac{3}{5} \div \left (\frac{1}{5}+2 \right )$

$= \frac{3}{5} \div \frac{11}{5}$

$= \frac{3}{5} \cdot \frac{5}{11}$

$= \frac{3}{1} \cdot \frac{1}{11}$

$= \frac{3} {11}$

Define an operation $\blacklozenge$ as follows:

For all real numbers ,

$a ; \blacklozenge ; b = a \div b^{2}$ .

Evaluate $25 ; \blacklozenge ; 6\frac{1}{4}$ .

$\frac{16} {25}$

$\frac{4}{25}$

$976\frac{9}{16}$

Explanation

$a ; \blacklozenge ; b = a \div b^{2}$

$25 ; \blacklozenge ; 6\frac{1}{4}= 25 \div \left ( 6\frac{1}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{25}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{625}{16} \right )$

$= \frac{25 }{1}\times \left ( \frac{16} {625}\right )$

$= \frac{1}{1}\times \left ( \frac{16} {25}\right )$

$= \frac{16} {25}$

Define an operation $\amalg$ as follows:

For all real numbers ,

$a \amalg b = 5 - \frac{a}{b}$ .

Evaluate $1 \frac{1}{2}; \amalg ; 2 \frac{1}{3}$ .

$4 \frac{5}{14}$

$1\frac{1}{2}$

$3\frac{4}{9}$

$4\frac{5}{7}$

The correct answer is not among the other responses.

Explanation

$a \amalg b = 5 - \frac{a}{b}$

or, equivalently,

$a \amalg b = 5 - a \div b$

$1 \frac{1}{2}; \amalg ; 2 \frac{1}{3}= 5 - 1 \frac{1}{2} \div 2 \frac{1}{3}$

$= 5 - \frac{3}{2} \div \frac{7}{3}$

$= 5 - \frac{3}{2} \times \frac{3}{7}$

$= 5 - \frac{9}{14}$

$= 4 \frac{5}{14}$

Define an operation $\ominus$ as follows:

For all real numbers ,

$a ; \ominus ; b = a^{2} \div b$ .

Evaluate $16 ; \ominus ; 3\frac{1}{5}$ .

$\frac{5}{16}$

$1\frac{9}{16}$

$819\frac{1}{5}$

Explanation

$a ; \ominus ; b = a^{2} \div b$

$16 ; \ominus ; 3\frac{1}{5} = 16^{2} \div 3\frac{1}{5}$

$= 256 \div \frac{16}{5}$

$= \frac{256 }{1}\times \frac{5}{16}$

$= \frac{16}{1}\times \frac{5}{1}$

$= 16 \times 5 = 80$

Simplify: $\frac{\frac{3x}{2}}{\frac{5x}{4}}$

$\frac{6}{5}$

$\frac{15x^{2}}{8}$

$\frac{15x}{8}$

$\frac{15x}{2}$

Explanation

Start by rewriting this fraction as a division problem:

$\frac{3x}{2}\div \frac{5x}{4}$

When dividing fractions, you multiply by the reciprocal of the second fraction, so you can rewrite your problem like this:

$\frac{3x}{2}\times \frac{4}{5x}$

Multiply across the numerators and then across the denominators to get $\frac{12x}{10x}$ . The x's cancel, and you can reduce the fraction to be $\frac{6}{5}$ .

Simplify:

$\frac{1+5}{\frac{6}{3+1}}$

$\frac{1}{4}$

$\frac{5}{8}$

$\frac{16}{3}$

Explanation

Begin by simplifying any additions that need to be done:

$\frac{1+5}{\frac{6}{3+1}}$

becomes

$\frac{6}{\frac{6}{4}}$

Now, remember that the numerator can be rewritten $\frac{6}{1}$ :

$\frac{\frac{6}{1}}{\frac{6}{4}}$

Now, when you divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{6}{1} * \frac{4}{6}$

Cancel the s and you get:

If $\dpi{100} \small x=\frac{2}{3}$ and $\dpi{100} \small y= \frac {3}{4}$ , then what is the value of $\dpi{100} \small \frac {x}{y}$ ?

$\dpi{100} \small \frac{8}{9}$

$\dpi{100} \small \frac{9}{8}$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small 2$

$\dpi{100} \small \frac{5}{7}$

Explanation

Dividing by a number (in this case $\dpi{100} \small \frac {3}{4}$ ) is equivalent to multiplying by its reciprocal (in this case $\dpi{100} \small \frac {4}{3}$ ). Therefore: