Solving Open Sentences with Arithmetic Operations for JSS 1

Use of Symbols (contd.)

Performance Objectives

By the end of the lesson, students should be able to:

1. solve open sentences problems involving two arithmetic operations;
2. solve word-problems involving use of symbols;
3. solve quantitative aptitude problems on the use of symbols.

Solving Open Sentences with Two Arithmetic Operations

An open sentence is a mathematical statement that contains an unknown value represented by a symbol or a letter.

The statement is called “open” because its truth depends on the value substituted for the symbol. For example, the sentence $x + 4 = 10$ is true if $x = 6$, but false for any other value of $x$.

Solving an open sentence means finding the exact value of the symbol that makes the mathematical statement true. When an open sentence contains two arithmetic operations, such as multiplication and addition, or division and subtraction, two distinct steps are required to isolate the symbol.

To solve these problems, inverse operations are performed in the reverse order of the standard order of operations. The addition or subtraction is undone first, followed by the multiplication or division.

Worked Example 1

Solve the open sentence:

$$3x + 5 = 17$$

The solution process involves the following steps:

1. Identify the two operations acting on the variable $x$, which are multiplication by $3$ and addition of $5$.
2. Undo the addition first by subtracting $5$ from both sides of the equation:
   $$3x = 17 – 5$$
   $$3x = 12$$
3. Undo the multiplication by dividing both sides of the equation by $3$:
   $$x = \frac{12}{3}$$
   $$x = 4$$

The value of the symbol $x$ is $4$.

Worked Example 2

Solve the open sentence:

$$\frac{y}{2} – 3 = 5$$

The solution process involves the following steps:

1. Identify the two operations acting on the variable $y$, which are division by $2$ and subtraction of $3$.
2. Undo the subtraction first by adding $3$ to both sides of the equation:
   $$\frac{y}{2} = 5 + 3$$
   $$\frac{y}{2} = 8$$
3. Undo the division by multiplying both sides of the equation by $2$:
   $$y = 8 \times 2$$
   $$y = 16$$

The value of the symbol $y$ is $16$.

Worked Example 3

Solve the open sentence below to find the value of the box symbol:

$$4\square – 7 = 13$$

The solution process involves the following steps:

1. Identify the operations acting on the box symbol, which are multiplication by $4$ and subtraction of $7$.
2. Undo the subtraction by adding $7$ to both sides of the equation:
   $$4\square = 13 + 7$$
   $$4\square = 20$$
3. Undo the multiplication by dividing both sides of the equation by $4$:
   $$\square = \frac{20}{4}$$
   $$\square = 5$$

The value of the box symbol is $5$.

Word Problems Involving the Use of Symbols

Word problems are mathematical problems expressed in everyday language. To solve these problems, real-world situations must be translated into mathematical open sentences using symbols or letters to represent the unknown quantities.

Polya’s Problem Solving Model provides a structured, four-stage approach to solving word problems systematically.

The stages of Polya’s Problem Solving Model are:

1. Understand the problem: Read the problem carefully to identify the given information and what needs to be found. Choose a letter or symbol to represent the unknown quantity.
2. Devise a plan: Translate the relationships described in the words into an algebraic open sentence.
3. Carry out the plan: Solve the open sentence using step-by-step algebraic methods.
4. Look back: Check the obtained solution against the original word problem to ensure the answer is correct and makes logical sense.

Worked Example 1

When a certain number is multiplied by $3$ and then $4$ is added, the result is $19$. Find the number.

The solution using Polya’s model involves the following stages:

1. Understand the problem: An unknown number is multiplied by $3$, then $4$ is added to get $19$. Let the unknown number be represented by the symbol $n$.
2. Devise a plan: Translate the words into an open sentence:
   $$3n + 4 = 19$$
3. Carry out the plan: Solve the open sentence:
   $$3n = 19 – 4$$
   $$3n = 15$$
   $$n = \frac{15}{3}$$
   $$n = 5$$
4. Look back: Substitute $5$ back into the original statement. Multiply $5$ by $3$ to get $15$, then add $4$ to get $19$. The solution is correct.

Worked Example 2

Amina has some books. She doubles the number of books and then gives $3$ to her brother. She is left with $11$ books. How many books did she have originally?

The solution using Polya’s model involves the following stages:

1. Understand the problem: Amina starts with an unknown number of books. Doubling means multiplying by $2$. Giving away $3$ means subtracting $3$. The final amount is $11$. Let the original number of books be represented by the symbol $b$.
2. Devise a plan: Translate the words into an open sentence:
   $$2b – 3 = 11$$
3. Carry out the plan: Solve the open sentence:
   $$2b = 11 + 3$$
   $$2b = 14$$
   $$b = \frac{14}{2}$$
   $$b = 7$$
4. Look back: If Amina started with $7$ books, doubling them gives $14$. Giving $3$ away leaves her with $11$. The solution is correct.

Worked Example 3

Chidi thought of a number. He divided the number by $4$ and then subtracted $6$ from the result. His final answer was $2$. What number did Chidi think of?

The solution using Polya’s model involves the following stages:

1. Understand the problem: An unknown number is divided by $4$, and then $6$ is subtracted to get $2$. Let the number Chidi thought of be represented by the symbol $y$.
2. Devise a plan: Translate the words into an open sentence:
   $$\frac{y}{4} – 6 = 2$$
3. Carry out the plan: Solve the open sentence:
   $$\frac{y}{4} = 2 + 6$$
   $$\frac{y}{4} = 8$$
   $$y = 8 \times 4$$
   $$y = 32$$
4. Look back: If the number is $32$, dividing it by $4$ gives $8$. Subtracting $6$ from $8$ gives $2$. The solution is correct.

Quantitative Aptitude Problems on the Use of Symbols

Quantitative aptitude problems test the ability to identify mathematical patterns, relationships, or custom rules defined by symbols, and apply those rules to find missing values.

In these problems, standard symbols like $\otimes$, $\oplus$, or shapes like triangles and boxes are given temporary mathematical definitions. The key is to analyze the given example, deduce the mathematical rule, and apply it to solve the target problem.

Worked Example 1

If the symbol $\otimes$ is defined by the rule $a \otimes b = 2a + b$, find the value of $4 \otimes 5$.

The solution process involves the following steps:

1. Identify the positions of the numbers relative to the symbol. Here, $a = 4$ and $b = 5$.
2. Substitute these values into the defined rule:
   $$4 \otimes 5 = 2(4) + 5$$
3. Calculate the final value:
   $$2(4) + 5 = 8 + 5 = 13$$

The value of $4 \otimes 5$ is $13$.

Worked Example 2

Study the relationship in the pattern below and find the missing value represented by the symbol $\Delta$.

The pattern is shown in three diagrams where a top number is related to two bottom numbers:

1. First diagram: Top number is $11$, bottom-left is $2$, bottom-right is $5$.
2. Second diagram: Top number is $19$, bottom-left is $3$, bottom-right is $6$.
3. Third diagram: Top number is $29$, bottom-left is $4$, bottom-right is $\Delta$.

The solution process involves the following steps:

1. Analyze the first diagram to find the rule:
   $$(2 \times 5) + 1 = 10 + 1 = 11$$
   The rule is: (Bottom-Left $\times$ Bottom-Right) $+ 1 =$ Top.
2. Test the rule on the second diagram to verify:
   $$(3 \times 6) + 1 = 18 + 1 = 19$$
   The rule is correct.
3. Apply the rule to the third diagram to find the value of $\Delta$:
   $$(4 \times \Delta) + 1 = 29$$
   $$4\Delta + 1 = 29$$
   $$4\Delta = 29 – 1$$
   $$4\Delta = 28$$
   $$\Delta = \frac{28}{4}$$
   $$\Delta = 7$$

The value of the symbol $\Delta$ is $7$.

Worked Example 3

If the operation $\star$ is defined as $x \star y = \frac{x + y}{2}$, find the value of $w$ in the open sentence:

$$8 \star w = 10$$

The solution process involves the following steps:

1. Apply the definition of the operation to the left side of the equation:
   $$8 \star w = \frac{8 + w}{2}$$
2. Set this expression equal to $10$:
   $$\frac{8 + w}{2} = 10$$
3. Solve the resulting open sentence for $w$:
   $$8 + w = 10 \times 2$$
   $$8 + w = 20$$
   $$w = 20 – 8$$
   $$w = 12$$

The value of $w$ is $12$.

Practice Exercises

The following exercises are provided for practice and evaluation:

1. Solve the following open sentences to find the value of the unknown letter:
   1. $5a + 3 = 18$
   2. $2y – 9 = 15$
   3. $\frac{m}{3} + 4 = 10$
   4. $\frac{k}{5} – 2 = 6$
2. Translate the following word problems into open sentences using a symbol of choice, and solve them:
   1. A certain number is multiplied by $6$, and then $5$ is subtracted from the product. If the final result is $25$, find the number.
   2. Olu thought of a number. He doubled it and added $8$. The result was $24$. What number did Olu think of?
   3. A teacher shared a pack of sweets among $3$ students. After each student received an equal share, the teacher added $2$ more sweets to one student’s share, making their total $10$ sweets. How many sweets were originally in the pack?
3. Solve the following quantitative aptitude problems:
   1. If $p \oplus q = 3p – q$, find the value of $5 \oplus 4$.
   2. If $x \odot y = (x + 2) \times y$, find the value of $a$ in the sentence $a \odot 3 = 15$.
   3. If the symbol $\square$ represents an operation such that $m \square n = \frac{m \times n}{3}$, find the value of $6 \square 9$.