How to subtract fractions - PSAT Math

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Question

Subtract and simplify the fractions:

$\small $\frac{7}{5}$ - $\frac{3}{6}$$

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Answer

First we need to give them common denominators. Remember, whatever you do to the denominator you must do to the numerator:

$\small \small $\frac{6}{6}$ \times $\frac{7}{5}$ -$\frac{5}{5}$ \times $\frac{3}{6}$$

$\small = $\frac{42}{30}$- $\frac{15}{30}$$

From here, we do algebraic operations to subtract and simplify.

$\small = $\frac{42 - 15}{30}$$

$\small = $\frac{27}{30}$$

$\small = $\frac{9}{10}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = $\frac{3}{5}$ a -b$

Evaluate $\frac{1}{3 }$\Cup $\frac{1}{10}$

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Answer

$a \Cup b = $\frac{3}{5}$ a -b$

$\frac{1}{3 }$\Cup $\frac{1}{10}$ = $\frac{3}{5}$\left ($\frac{1}{3}$ \right ) - $\frac{1}{10}$

$= $\frac{1}{5}$\left ($\frac{1}{1}$ \right ) - $\frac{1}{10}$$

$= $\frac{1}{5}$ - $\frac{1}{10}$$

$= $\frac{2}{10}$ - $\frac{1}{10}$$

$= $\frac{1}{10}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = 2a - 3b$ .

Evaluate $\frac{4}{5}$ \bigstar $\frac{3}{10}$ .

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Answer

$a \bigstar b = 2a - 3b$

$\frac{4}{5}$ \bigstar $\frac{3}{10}$ = 2 \cdot $\frac{4}{5}$ - 3 \cdot $\frac{3}{10}$

$= $\frac{2}{1}$ \cdot $\frac{4}{5}$ - $\frac{3}{1}$ \cdot $\frac{3}{10}$$

$= $\frac{8}{5}$ - $\frac{9}{10}$$

$= $\frac{16}{10}$ - $\frac{9}{10}$$

$= $\frac{7}{10}$$

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Question

Define an operation $\triangleright$ as follows:

For all real ,

$a \triangleright b = 7 - ab$ .

Evaluate $5 \triangleright \left ( - $\frac{5}{3}\right )$ .

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Answer

$a \triangleright b = 7 - ab$

$5 \triangleright \left ( - $\frac{5}{3}\right) = 7 - 5 \left( - $\frac{5}{3}\right )$

$= 7 - $\frac{5}{1}$ \left ( - $\frac{5}{3}\right)$

$= 7 - \left( - $\frac{25}{3}\right )$

$= $\frac{21}{3}$ + $\frac{25}{3}$$

$= $\frac{46}{3}$$

$=15 $\frac{1}{3}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = a - $b^{2}$$

Evaluate $5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$$ .

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Answer

$a \Cup b = a - $b^{2}$$

$5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$ = 5$\frac{1}{2}$ -\left ( 1 $\frac{1}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $\frac{4}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $$\frac{4^{2}$$$}{3^{2}$} \right)$

$= $\frac{11}{2}$ - $\frac{16}{9}$$

$= $\frac{99}{18}$ - $\frac{32}{18}$$

$= $\frac{67}{18}$$

$=3 $\frac{13}{18}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$ .

Evaluate $15 \bigstar 10$ .

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Answer

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$

$15 \bigstar 10 = $\frac{1}{2}$ \cdot 15 - $\frac{1}{3}$ \cdot 10$

$= $\frac{1}{2}$ \cdot $\frac{15 }{1}$- $\frac{1}{3}$ \cdot $\frac{10}{1}$$

$= $\frac{15}{2}$ - $\frac{10}{3}$$

$= $\frac{45}{6}$ - $\frac{20}{6}$$

$= $\frac{25}{6}$$

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

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Question

Define an operation $\triangleright$ as follows:

For all real ,

$a \triangleright b = $a^{2}$- $b^{3}$$ .

Evaluate $\frac{4}{5}$ \triangleright $\frac{4}{5}$ .

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Answer

$a \triangleright b = $a^{2}$- $b^{3}$$

$\frac{4}{5}$ \triangleright $\frac{4}{5}$ =\left ( $\frac{4}{5}$ \right $)^{2}$ - \left ( $\frac{4}{5}$ \right $)^{3}$

$=\left ( $$\frac{4^{2}$$$}{5^{2}$} \right) - \left( $$\frac{4^{3}$$$}{5^{3}$} \right )$

$= $\frac{16}{25}$ - $\frac{64}{125}$$

$= $\frac{80}{125}$ - $\frac{64}{125}$$

$= $\frac{16}{125}$$

This answer is not among the choices given.

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Question

Subtract and simplify the fractions:

$\small $\frac{7}{5}$ - $\frac{3}{6}$$

Tap to reveal answer

Answer

First we need to give them common denominators. Remember, whatever you do to the denominator you must do to the numerator:

$\small \small $\frac{6}{6}$ \times $\frac{7}{5}$ -$\frac{5}{5}$ \times $\frac{3}{6}$$

$\small = $\frac{42}{30}$- $\frac{15}{30}$$

From here, we do algebraic operations to subtract and simplify.

$\small = $\frac{42 - 15}{30}$$

$\small = $\frac{27}{30}$$

$\small = $\frac{9}{10}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = $\frac{3}{5}$ a -b$

Evaluate $\frac{1}{3 }$\Cup $\frac{1}{10}$

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Answer

$a \Cup b = $\frac{3}{5}$ a -b$

$\frac{1}{3 }$\Cup $\frac{1}{10}$ = $\frac{3}{5}$\left ($\frac{1}{3}$ \right ) - $\frac{1}{10}$

$= $\frac{1}{5}$\left ($\frac{1}{1}$ \right ) - $\frac{1}{10}$$

$= $\frac{1}{5}$ - $\frac{1}{10}$$

$= $\frac{2}{10}$ - $\frac{1}{10}$$

$= $\frac{1}{10}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = 2a - 3b$ .

Evaluate $\frac{4}{5}$ \bigstar $\frac{3}{10}$ .

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Answer

$a \bigstar b = 2a - 3b$

$\frac{4}{5}$ \bigstar $\frac{3}{10}$ = 2 \cdot $\frac{4}{5}$ - 3 \cdot $\frac{3}{10}$

$= $\frac{2}{1}$ \cdot $\frac{4}{5}$ - $\frac{3}{1}$ \cdot $\frac{3}{10}$$

$= $\frac{8}{5}$ - $\frac{9}{10}$$

$= $\frac{16}{10}$ - $\frac{9}{10}$$

$= $\frac{7}{10}$$

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Question

Define an operation $\triangleright$ as follows:

For all real ,

$a \triangleright b = 7 - ab$ .

Evaluate $5 \triangleright \left ( - $\frac{5}{3}\right )$ .

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Answer

$a \triangleright b = 7 - ab$

$5 \triangleright \left ( - $\frac{5}{3}\right) = 7 - 5 \left( - $\frac{5}{3}\right )$

$= 7 - $\frac{5}{1}$ \left ( - $\frac{5}{3}\right)$

$= 7 - \left( - $\frac{25}{3}\right )$

$= $\frac{21}{3}$ + $\frac{25}{3}$$

$= $\frac{46}{3}$$

$=15 $\frac{1}{3}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = a - $b^{2}$$

Evaluate $5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$$ .

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Answer

$a \Cup b = a - $b^{2}$$

$5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$ = 5$\frac{1}{2}$ -\left ( 1 $\frac{1}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $\frac{4}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $$\frac{4^{2}$$$}{3^{2}$} \right)$

$= $\frac{11}{2}$ - $\frac{16}{9}$$

$= $\frac{99}{18}$ - $\frac{32}{18}$$

$= $\frac{67}{18}$$

$=3 $\frac{13}{18}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$ .

Evaluate $15 \bigstar 10$ .

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Answer

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$

$15 \bigstar 10 = $\frac{1}{2}$ \cdot 15 - $\frac{1}{3}$ \cdot 10$

$= $\frac{1}{2}$ \cdot $\frac{15 }{1}$- $\frac{1}{3}$ \cdot $\frac{10}{1}$$

$= $\frac{15}{2}$ - $\frac{10}{3}$$

$= $\frac{45}{6}$ - $\frac{20}{6}$$

$= $\frac{25}{6}$$

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

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Question

Define an operation $\triangleright$ as follows:

For all real ,

$a \triangleright b = $a^{2}$- $b^{3}$$ .

Evaluate $\frac{4}{5}$ \triangleright $\frac{4}{5}$ .

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Answer

$a \triangleright b = $a^{2}$- $b^{3}$$

$\frac{4}{5}$ \triangleright $\frac{4}{5}$ =\left ( $\frac{4}{5}$ \right $)^{2}$ - \left ( $\frac{4}{5}$ \right $)^{3}$

$=\left ( $$\frac{4^{2}$$$}{5^{2}$} \right) - \left( $$\frac{4^{3}$$$}{5^{3}$} \right )$

$= $\frac{16}{25}$ - $\frac{64}{125}$$

$= $\frac{80}{125}$ - $\frac{64}{125}$$

$= $\frac{16}{125}$$

This answer is not among the choices given.

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Question

Subtract and simplify the fractions:

$\small $\frac{7}{5}$ - $\frac{3}{6}$$

Tap to reveal answer

Answer

First we need to give them common denominators. Remember, whatever you do to the denominator you must do to the numerator:

$\small \small $\frac{6}{6}$ \times $\frac{7}{5}$ -$\frac{5}{5}$ \times $\frac{3}{6}$$

$\small = $\frac{42}{30}$- $\frac{15}{30}$$

From here, we do algebraic operations to subtract and simplify.

$\small = $\frac{42 - 15}{30}$$

$\small = $\frac{27}{30}$$

$\small = $\frac{9}{10}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = $\frac{3}{5}$ a -b$

Evaluate $\frac{1}{3 }$\Cup $\frac{1}{10}$

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Answer

$a \Cup b = $\frac{3}{5}$ a -b$

$\frac{1}{3 }$\Cup $\frac{1}{10}$ = $\frac{3}{5}$\left ($\frac{1}{3}$ \right ) - $\frac{1}{10}$

$= $\frac{1}{5}$\left ($\frac{1}{1}$ \right ) - $\frac{1}{10}$$

$= $\frac{1}{5}$ - $\frac{1}{10}$$

$= $\frac{2}{10}$ - $\frac{1}{10}$$

$= $\frac{1}{10}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = 2a - 3b$ .

Evaluate $\frac{4}{5}$ \bigstar $\frac{3}{10}$ .

Tap to reveal answer

Answer

$a \bigstar b = 2a - 3b$

$\frac{4}{5}$ \bigstar $\frac{3}{10}$ = 2 \cdot $\frac{4}{5}$ - 3 \cdot $\frac{3}{10}$

$= $\frac{2}{1}$ \cdot $\frac{4}{5}$ - $\frac{3}{1}$ \cdot $\frac{3}{10}$$

$= $\frac{8}{5}$ - $\frac{9}{10}$$

$= $\frac{16}{10}$ - $\frac{9}{10}$$

$= $\frac{7}{10}$$

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Question

Define an operation $\triangleright$ as follows:

For all real ,

$a \triangleright b = 7 - ab$ .

Evaluate $5 \triangleright \left ( - $\frac{5}{3}\right )$ .

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Answer

$a \triangleright b = 7 - ab$

$5 \triangleright \left ( - $\frac{5}{3}\right) = 7 - 5 \left( - $\frac{5}{3}\right )$

$= 7 - $\frac{5}{1}$ \left ( - $\frac{5}{3}\right)$

$= 7 - \left( - $\frac{25}{3}\right )$

$= $\frac{21}{3}$ + $\frac{25}{3}$$

$= $\frac{46}{3}$$

$=15 $\frac{1}{3}$$

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Question

Define an operation $\Cup$ as follows:

For all real ,

$a \Cup b = a - $b^{2}$$

Evaluate $5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$$ .

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Answer

$a \Cup b = a - $b^{2}$$

$5$\frac{1}{2}$ \Cup 1 $\frac{1}{3}$ = 5$\frac{1}{2}$ -\left ( 1 $\frac{1}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $\frac{4}{3}$ \right $)^{2}$$

$= $\frac{11}{2}$ -\left ( $$\frac{4^{2}$$$}{3^{2}$} \right)$

$= $\frac{11}{2}$ - $\frac{16}{9}$$

$= $\frac{99}{18}$ - $\frac{32}{18}$$

$= $\frac{67}{18}$$

$=3 $\frac{13}{18}$$

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Question

Define an operation $\bigstar$ as follows:

For all real ,

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$ .

Evaluate $15 \bigstar 10$ .

Tap to reveal answer

Answer

$a \bigstar b = $\frac{1}{2}$ a - $\frac{1}{3}$ b$

$15 \bigstar 10 = $\frac{1}{2}$ \cdot 15 - $\frac{1}{3}$ \cdot 10$

$= $\frac{1}{2}$ \cdot $\frac{15 }{1}$- $\frac{1}{3}$ \cdot $\frac{10}{1}$$

$= $\frac{15}{2}$ - $\frac{10}{3}$$

$= $\frac{45}{6}$ - $\frac{20}{6}$$

$= $\frac{25}{6}$$

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

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