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# Chapter 9: Using symbols

## 9.1 Open and closed number sentences

### Closed number sentences

A number sentence is a group of numbers and mathematical symbols (such as $+$, $-$ or $=$) that gives information about the numbers. Here is an example of a number sentence:

This is called a closed number sentence, because all the information is given in the sentence. This closed number sentence is always true.

Here is an example of a closed number sentence that is false:

closed number sentence A closed number sentence gives all the information about the numbers, and it is always true or always false.

Here are some more examples of closed number sentences:

$9 \div 3 = 3$ (true)

$7\times 12 = 84$ (true)

$4 \times 8 = 8 + 8 + 8 + 8 + 8$ (false)

### Exercise 9.1: Decide whether closed sentences are true or false

State whether the closed sentence is always true or false.

1. True

2. False

3. True

4. False

5. True

### Open number sentences

Here is another example of a number sentence:

This is called an open number sentence, because not all the information is given in the sentence. We do not know if this sentence is true or false because we do not know what is in the box.

open number sentence An open number sentence does not give all the information about the numbers, but has information missing that we must find.

### Worked example 9.1: Finding missing information in an open sentence

Find the value of the square in this open number sentence:

1. Step 1: Ask yourself the question, "What do I need to add to 3 to get 7?"

Using your knowledge and mental maths, you will find that you need to add 4 to 3 to get 7.

2. Step 2: Write down the value of the square.

### Exercise 9.2: Find missing information in open sentences

For each open number sentence, find the number that makes the sentence true.

1. 8

2. 12

3. 24

4. 7

5. 1

6. 10

7. 7

8. 30

9. 5

10. 7

### Variables in open sentences

So far we have worked with open sentences that contain a symbol such as a $\square$ or a $\bigcirc$. Instead of pictures like these, in mathematics it works better to use letters to represent missing information. These letters are called variables, because their value can vary.

variable A variable is a symbol for a number we do not know yet. It is usually a letter such as $x$ or $y$.

If we have an expression, such as $x+5$, we cannot know the value of the $x$ unless it is given. If we give the $x$ different values, then we can make number sentences that are true.

If $x$ = 1, then $x+5 = 1 + 5 = 6$

If $x$ = 3, then $x+5 = 3 + 5 = 8$

If $x$ = 0, then $x+5 = 0 + 5 = 0$

The value of the $x$ can vary, and that is why it is called a variable.

### Exercise 9.3: Find the value of the expression

Find the value of the following expressions if $x=3$.

1. 14

2. 6

3. 0

4. 8

5. 30

6. 20

7. 63

8. 18

9. 26

## 9.2 Solving open sentences that have variables

Variables and open sentences form part of a section of mathematics we call algebra. To show that there is information that we must find, an open sentence can use a picture symbol (such as a box or a circle) or a variable (a letter representing any number).

Here are some examples of open sentences using variables:

In these open sentences, we can find the correct value for the unknown variable that will make the sentence true. This is called solving the open sentence.

If the sentence that you need to be solve is straightforward, one option is to use mental maths to solve it.

### Worked example 9.2: Finding the value of a variable using mental maths

Find the value of $x$ in this open number sentence:

1. Step 1: You need to find the value of $x$ that will make the statement true. To do this, you can ask yourself the question, "What number added to 7 will give me 9?"

Using your knowledge and mental maths, you will find that you need to add 2 to 7 to get 9.

2. Step 2: This means that the value of $x$ is 2, so write your answer:

### Using inverse operations to solve open sentences

Another way to find the value of $x$ in $x + 7 = 9$, is to use the inverse operation of addition. The inverse operation of addition is subtraction. In Chapter 5 you learnt that subtraction may be seen as another form of addition. For example, $6-4$ is the same as $6+(-4)$. The numbers $4$ and $-4$ are called additive inverses. When we add them, the answer is zero: $4+(-4)=0$.

The inverse operation of multiplication is division. In Chapter 6 you learnt that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. For example, $4\div\frac{2}{3}$ is the same as $4\times\frac{3}{2}$. A fraction and its reciprocal are called multiplicative inverses. When we multiply them, the answer is 1: $\frac{2}{3}\times\frac{3}{2}=1$.

inverse operation An inverse operation does the opposite of another operation. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.

This closed number sentence is true:

We know that this closed number sentence is also true:

Notice that we get from the first number sentence to the second number sentence using the inverse operation of addition, which is subtraction:

\begin{align} 2 + 7 &= 9 \newline 2 + 7\, \color{red}{-\;7} &= 9\space \color{red}{-\;7} \newline 2 + 0 &= 9 - 7 \newline 2 &= 9 - 7 \end{align}

Here is an example of a closed number sentence using multiplication:

We know that this closed number sentence is also true:

Notice that we get from the first number sentence to the second number sentence using the inverse operation of multiplication, which is division:

\begin{align} 2 \times 3 &= 6 \newline 2 \times 3\; \color{red}{\div\;3} &= 6\space \color{red}{\div\;3} \newline 2 \times 1 &= 6 \div 3 \newline 2 &= 6 \div 3 \end{align}

We may use inverse operations to find the value of a variable in an open number sentence. We aim to write the number sentence in this format:

### Worked example 9.3: Solving an open sentence using the inverse of addition

Find the value of $x$ in this open sentence:

1. Step 1: Use an inverse operation for any number that is added to or subtracted from the variable.

2. Step 2: Simplify the sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$.

\begin{align} x + 12 - 12 &= 29 - 12 \newline x + 0 &= 17 \newline x &= 17 \end{align}

### Worked example 9.4: Solving an open sentence using the inverse of multiplication

Find the value of $y$ in this open sentence:

1. Step 1: Use an inverse operation for any number that the variable is multiplied or divided by.

2. Step 2: Simplify the number sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$.

\begin{align} y \times 3 \div 3 &= 27 \div 3 \newline y \times 1 &= 9 \newline y &= 9 \end{align}

Note that an open sentence is not regarded as solved when the variable is negative, for example $-x=5$. The variable must be positive in your final answer, for example $x=-5$.

The format $variable=\text{value}$ may be swapped around. Therefore, $x=7$ and $7=x$ has the same meaning.

### Worked example 9.5: Solving an open sentence using the inverse operation on the variable

Find the value of $x$:

1. Step 1: Use an inverse operation to get a positive variable.

2. Step 2: Simplify the number sentence from Step 1 until it has only one positive variable.

\begin{align} 18 -x + x &= 14 + x \newline 18 + 0 &= 14 + x \newline 18 &= 14 + x \end{align}
3. Step 3: Use an inverse operation for the number that is added to or subtracted from the variable.

4. Step 4: Simplify the number sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$.

\begin{align} 18 - 14 &= 14 -14 + x \newline 4 &= 0 + x \newline 4 &= x \newline x &= 4 \end{align}

### Exercise 9.4: Solve open sentences

Find the value of the variable in the following open sentences:

1. \begin{align} y + 7 - 7 &= 30 - 7 \newline y &= 23 \end{align}
2. \begin{align} 3 - 3 + x &= 15 - 3 \newline x &= 12 \end{align}
3. \begin{align} z - 8 + 8 &= 20 + 8 \newline z &= 28 \end{align}
4. \begin{align} q - 4 + 4 &= -5 + 4 \newline q &= -1 \end{align}
5. \begin{align} 17 - n + n &= 20 + n \newline 17 &= 20 + n \newline 17 - 20 &= 20 - 20 + n \newline 3 &= n \newline n &= 3 \end{align}
6. \begin{align} 15 - p + p &= -5 + p \newline 15 &= -5 + p \newline 15 + 5 &= -5 + 5 + p \newline 20 &= p \newline p &= 20 \end{align}
7. \begin{align} x \times 8 \div 8 &= 48 \div 8 \newline x &= 6 \newline \end{align}
8. \begin{align} 3 \div 3 \times y &= 15 \div 3 \newline y &= 5 \newline \end{align}
9. \begin{align} c \div 20 \times 20 &= 5 \times 20 \newline c &= 100 \newline \end{align}
10. \begin{align} 40 \div x \times x &= 8 \times x \newline 40 &= 8 \times x \newline 40 \div 8 &= 8 \div 8 \times x \newline 5 &= x \newline x &= 5 \end{align}

### Solve open sentences with two operations

Open number sentences may have more than one operation, for example $4-2+x=10$. To solve number sentences like these, you must remember the correct order of operations that you learnt last year:

1. Brackets
2. Of
3. Divide
4. Multiply
6. Subtract

### Worked example 9.6: Solving open sentences with two operations

Find the value of $x$:

1. Step 1: Decide which inverse operation to do first.

According to the order BODMAS, you must calculate $x \div 2$ before $2+3$.
This means $x \div 2$ must "stay together". You may insert brackets to help you.
We must use the inverse of addition first.

2. Step 2: Simplify the number sentence from Step 1 until you have only one operation.

\begin{align} (x \div 2) +3 - 3 &= 8 - 3 \newline x \div 2 &= 5 \end{align}
3. Step 3: Use an inverse operation and simplify to get to the format $\boldsymbol{variable}=\textbf{ value}$.

\begin{align} x \div 2 &= 5 \newline x \div 2 \times 2 &= 5 \times 2 \newline x &= 10 \end{align}

### Exercise 9.5: Solve open sentences with two operations

Find the value of the variable in the following open sentences:

1. \begin{align} x - 1 &= 20 \newline x - 1 + 1 &= 20 + 1 \newline x &= 21 \end{align}
2. \begin{align} (q \times 3) + 5 - 5 &= 29 - 5 \newline q \times 3 &= 24 \newline q \times 3 \div 3 &= 24 \div 3 \newline q &= 8 \end{align}
3. \begin{align} t + (10 \div 2) &= 12 \newline t + 5 &= 12 \newline t + 5 - 5 &= 12 - 5 \newline t &= 7 \end{align}
4. \begin{align} 5 + 2 - x &= 12 \newline 7 - x &= 12 \newline 7 - x + x &= 12 + x \newline 7 &= 12 + x \newline 7 - 12 &= 12 - 12 + x \newline -5 &= x \newline x &= -5 \end{align}
5. \begin{align} (y \times 5) - 7 &= 13 \newline (y \times 5) - 7 + 7 &= 13 + 7 \newline y \times 5 &= 20 \newline y \times 5 \div 5 &= 20 \div 5 \newline y &= 4 \end{align}
6. \begin{align} (p \div 2) - 5 &= 25 \newline (p \div 2) - 5 + 5 &= 25 + 5 \newline p \div 2 &= 30 \newline p \div 2 \times 2 &= 30 \times 2 \newline p &= 60 \end{align}
7. \begin{align} -10 &= 6 + v \newline -10 - 6 &= 6 - 6 + v \newline -16 &= v \newline v &= -16 \end{align}
8. \begin{align} -12 &= 24 \div r \newline -12 \times r &= 24 \div r \times r \newline -12 \div -12 \times r &= 24 \div -12 \newline r &= -2 \end{align}

## 9.3 Using open sentences to solve problems

We can use open sentences with variables to set out and solve problems that are described in words. Remember that we can translate the following words into mathematical operations:

English Mathematical symbol
sum of +
difference $-$
product $\times$
quotient $\div$

When you need to solve a problem that is described in words, follow these steps:

1. Understand the problem: Read the question carefully a few times. Identify the information you have. Make sure you know what is asked. It may help to draw a picture.
2. Make a plan: Write a number sentence that you can use to solve the problem.
3. Carry out the plan: Solve the number sentence.

### Worked example 9.7: Solving a problem using open sentences

There are 32 students in one JSS1 class. If 20 of the students are boys, how many girls are there in this class?

1. Step 1: Understand the problem.

• total students: 32
• boys: 20
2. Step 2: Devise a plan.

\begin{align} \text{total students} &= \text{girls} + \text{boys}\newline 32 &= g + 20 \end{align}
3. Step 3: Carry out the plan.

\begin{align} 32 &= g + 20 \newline 32 - 20 &= g + 20 - 20 \newline 12 &= g \newline g &= 12 \end{align}
4. Step 4: Look back.

$12+20=32$
The answer is correct. There are 12 girls in the class.

### Worked example 9.8: Solving a problem using open sentences

For six days of the week, Ndidi spends the same amount of time each day practising the violin. On the last day of the week, she practises only 30 minutes. Her total practice time for the week is 300 minutes. For how many minutes does Ndidi practise the violin each day of the first six weekdays?

1. Step 1: Understand the problem.

• total time: 300 minutes
• last day: 30 minutes
• six days: equal minutes
2. Step 2: Devise a plan.

\begin{align} \text{total time} &= 6 \times \text{asked} + 30\newline 300 &= 6 \times x + 30 \end{align}
3. Step 3: Carry out the plan.

\begin{align} 300 &= (6 \times x) + 30 \newline 300 - 30 &= (6 \times x) + 30 - 30 \newline 270 &= 6 \times x \newline 270 \div 6 &= 6 \div 6 \times x \newline 45 &= x \newline x &= 45 \end{align}
4. Step 4: Look back.

$45 \text{ minutes} \times 6 \text{ days} = 270 \text{ minutes}$
$270 \text{ minutes} + 30 \text{ minutes} = 300 \text{ minutes}$
The answer is correct. She practises for 45 minutes per day for the first six days of the week.

### Exercise 9.6: Use open sentences to solve problems

1. Two numbers add up to 28. One of the numbers is 10. What is the other number?

\begin{align} x + 10 &= 28 \newline x &= 28 - 10 \newline x &= 18 \end{align}
2. The product of two numbers is 840. One of the numbers is 12. What is the other number?

\begin{align} x \times 12 &= 240 \newline x &= 240 \div 12 \newline x &= 20 \end{align}
3. 1,025 divided by a certain number gives a quotient (answer after dividing) of 25. What is the other number?

\begin{align} 1,025 \div x &= 25 \newline 1,025 &= 25 \times x \newline 1,025 \div 25 &= x \newline x &= 41 \end{align}
4. The difference between two numbers is 17. If the larger number is 67, what is the smaller number?

\begin{align} 67 - x &= 17 \newline 67 - 17 &= x \newline x &= 50 \end{align}
5. Usman is 1.8 m tall and his sister Jummai is $y$ m tall. The difference in their heights is 0.4 m. How tall is Jummai if she is shorter than Usman?

\begin{align} 1.8 - y &= 0.4 \newline 1.8 &= 0.4 + y \newline 1.8 - 0.4 &= y \newline y &= 1.4 \end{align}

Jummai is 1.4 m tall.

6. A farmer has 24 rows of apple trees in his apple orchard. Each row has the same number of apple trees. If he has a total of 480 apple trees, how many trees are there in each row?

\begin{align} 24 \times t &= 480 \newline t &= 480 \div 24 \newline t &= 20 \end{align}

There are 20 trees in each row.

7. The sum of three numbers is 208. One of the numbers is 88 and the other number is half of 88. What is the third number?

\begin{align} 88 + 44 + x &= 208 \newline x &= 208 - 132 \newline x &= 76 \end{align}
8. Two sisters add their ages and see that they are 100 years old together. The younger sister is 45. How old is her older sister?

\begin{align} 45 + x &= 100 \newline x &= 100 - 45 \newline x &= 55 \end{align}

The older sister is 55.

9. A mother has 4 children. She wants to divide 35 sweets equally among them. If every child receives 8 sweets, how many are left over?

\begin{align} 4 \times 8 + x &= 35 \newline 32 + x &= 35 \newline 32 - 32 + x &= 35 - 32 \newline x &= 3 \end{align}

There are 3 sweets left over.

10. There are 126 plates of food available at a party. There are 38 guests at the party and they each receive the same amount of food. The family who hosts the party gets 12 plates of food. How many plates of food does each guest get?

\begin{align} 38 \times x + 12 &= 126 \newline (38 \times x) + 12 - 12 &= 126 - 12 \newline 38 \times x &= 114 \newline 38 \div 38 \times x &= 114 \div 38 \newline x &= 3 \end{align}

Each guest gets 3 plates of food.

## 9.4 Summary

• A closed number sentence is always true or always false.
• In an open number sentence, there is some information missing that we need to find.
• An open number sentence can use picture symbols or variables to show which information is missing.
• The value of a variable can vary.
• Finding the missing information, or finding the value of the variable, is called solving an open sentence.
• We can solve an open sentence using mental maths or using the inverse operation.
• We can use open sentences to solve problems described in words.