Solve Multi-Step Problems with Positive and Negative Rational Numbers: CCSS.Math.Content.7.EE.B.3 - 7th Grade Math

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Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ .

$a \bigstar (-1) = 1$ . What is the value of ?

Answer

$a \bigstar b = \frac{a+b}{ab}$

and

$a \bigstar (-1) = 1$

so

$\frac{a+(-1)}{a(-1)}=1$

$\frac{a-1}{-a} = 1$

$\frac{a-1}{-a} \cdot (-a) = 1 \cdot (-a)$

$2a \div 2 = 1 \div 2$

$a = \frac{1}{2}$

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Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = a^{2}+ b$

If $6 \bigstar b = 100$ , then what is ?

Answer

$a \bigstar b = a^{2}+ b$ ,

so

$6 \bigstar b = 6 ^{2} + b = 36 + b$ .

If

$6 \bigstar b = 100$

then

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Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ .

If $a \bigstar \frac{1}{2} = -1$ , evaluate .

Answer

$a \bigstar b = \frac{a+b}{ab}$

so

$a \bigstar \frac{1}{2} = \frac{a+ \frac{1}{2} }{a \cdot \frac{1}{2} } = \frac{\frac{2a}{2}+ \frac{1}{2} }{ \frac{a}{2} } = \frac{\frac{2a+1}{2} }{ \frac{a}{2} }$

Equivalently

$a \bigstar \frac{1}{2} = \frac{2a+1}{2} \div \frac{a}{2} = \frac{2a+1}{2} \cdot \frac{2}{a} = \frac{2a+1}{a}$

$a \bigstar \frac{1}{2} = -1$ , so set this equal to :

$\frac{2a+1}{a} = -1$

$\frac{2a+1}{a} \cdot a = -1 \cdot a$

$3a\div 3 = - 1 \div 3$

$a = - \frac{1}{3}$

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Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ .

If $a \bigstar 0.5 = 0.1$ , then evaluate .

Answer

$a \bigstar b = \frac{a+b}{ab}$ ,

so

$a \bigstar 0.5 = \frac{a+0.5 }{a \cdot 0.5 } = \frac{a+0.5 }{0.5a }$

Set this equal to 0.1:

$\frac{a+0.5 }{0.5a } = 0.1$

$\frac{a+0.5 }{0.5a } \cdot 0.5 a = 0.1 \cdot 0.5 a$

$(a+0.5) \cdot 100 = 0.05 a \cdot 100$

$a \cdot 100 +0.5 \cdot 100 = 5 a$

$50 \div (-95) = -95a \div (-95)$

$a = -\frac{50}{95} = - \frac{10}{19}$

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Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{ab}{a+b}$

$a \bigstar 1= -1$ . What is the value of ?

Answer

$a \bigstar b = \frac{ab}{a+b}$ , so

$a \bigstar 1 = \frac{a \cdot 1 }{a+1} = \frac{a }{a+1}$

Set this equal to :

$\frac{a }{a+1} = -1$

$\frac{a }{a+1} \cdot (a+1) = -1 \cdot (a+1)$

$2a \div 2= - 1 \div 2$

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Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

We know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}25\ \times\ .20\end{array}}{ \ \ \space 5.00}$

The server's tip will be

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Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}29\ \times\ .20\end{array}}{ \ \ \space 5.80}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}17\ \times\ .20\end{array}}{ \ \ \space 3.40}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}33\ \times\ .20\end{array}}{ \ \ \space 6.60}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}35\ \times\ .20\end{array}}{ \ \ \space 7.00}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$20%\rightarrow0.20$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}46\ \times\ .20\end{array}}{ \ \ \space 9.20}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}25\ \times\ .15\end{array}}{ \ \ \space 3.75}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

We know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}15\ \times\ .15\end{array}}{ \ \ \space 2.25}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}38\ \times\ .15\end{array}}{ \ \ \space 5.70}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}42\ \times\ .15\end{array}}{ \ \ \space6.30}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}50\ \times\ .15\end{array}}{ \ \ \space 7.50}$

The server's tip will be

Compare your answer with the correct one above

Question

If a server at a restaurant gets a tip on a bill of , then what will be the amount of the tip?

Answer

The first step to solving this problem is determining what operation we are going to use.

Think:we know that the tip is going to be of . The keyword "of" gives us a clue and indicates that we are going to multiply.

As we have learned in previous lessons, we need to convert our percentage into a decimal before we are able to multiply:

$15%\rightarrow0.15$

Next, we can perform the multiplication operation and find the tip:

$\frac{\begin{array}[b]{r}57\ \times\ .15\end{array}}{ \ \ \space 8.55}$

The server's tip will be

Compare your answer with the correct one above

Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ .

$a \bigstar (-1) = 1$ . What is the value of ?

Answer

$a \bigstar b = \frac{a+b}{ab}$

and

$a \bigstar (-1) = 1$

so

$\frac{a+(-1)}{a(-1)}=1$

$\frac{a-1}{-a} = 1$

$\frac{a-1}{-a} \cdot (-a) = 1 \cdot (-a)$

$2a \div 2 = 1 \div 2$

$a = \frac{1}{2}$

Compare your answer with the correct one above

Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = a^{2}+ b$

If $6 \bigstar b = 100$ , then what is ?

Answer

$a \bigstar b = a^{2}+ b$ ,

so

$6 \bigstar b = 6 ^{2} + b = 36 + b$ .

If

$6 \bigstar b = 100$

then

Compare your answer with the correct one above

Question

Define an operation $\bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ .

If $a \bigstar \frac{1}{2} = -1$ , evaluate .

Answer

$a \bigstar b = \frac{a+b}{ab}$

so

$a \bigstar \frac{1}{2} = \frac{a+ \frac{1}{2} }{a \cdot \frac{1}{2} } = \frac{\frac{2a}{2}+ \frac{1}{2} }{ \frac{a}{2} } = \frac{\frac{2a+1}{2} }{ \frac{a}{2} }$

Equivalently

$a \bigstar \frac{1}{2} = \frac{2a+1}{2} \div \frac{a}{2} = \frac{2a+1}{2} \cdot \frac{2}{a} = \frac{2a+1}{a}$

$a \bigstar \frac{1}{2} = -1$ , so set this equal to :

$\frac{2a+1}{a} = -1$

$\frac{2a+1}{a} \cdot a = -1 \cdot a$

$3a\div 3 = - 1 \div 3$

$a = - \frac{1}{3}$

Compare your answer with the correct one above

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