Problem 40 - Prealgebra

Question

Simplify. $$\frac{5}{9}(77-32)$$

Step-by-Step Solution

Answer

The simplified result is 25.

Step 1: Simplify the expression inside the parentheses

Start by simplifying the expression inside the parentheses: \(77 - 32\). Subtract 32 from 77 to find the result of the expression inside the parentheses.

Step 2: Calculate the subtraction

Perform the subtraction: \(77 - 32 = 45\). Now the expression becomes \(\frac{5}{9} \times 45\).

Step 3: Simplify the multiplication

Multiply \(45\) by \(\frac{5}{9}\). This can be done by multiplying the numerator by 45 and then dividing by 9: \(\frac{5 \times 45}{9}\).

Step 4: Calculate the multiplication by simplifying

Now calculate \(5 \times 45 = 225\). The expression becomes \(\frac{225}{9}\).

Step 5: Divide the result

Finally, divide 225 by 9. Divide: \(225 \div 9 = 25\). Therefore, the simplified result is 25.

Key Concepts

Order of Operations

The order of operations is a fundamental principle in mathematics. It helps ensure that mathematical expressions are simplified or solved correctly. The basic idea is that there is a certain sequence in which calculations must be performed. This sequence is often remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• First, perform any calculations inside parentheses. This makes sure that you handle grouped expressions first. In our exercise, the operation inside the first set of parentheses, i.e., 77 - 32, was simplified first.
• Next, look for any exponents—but our current problem doesn't have any.
• After exponents, perform any multiplication or division as they appear from left to right. Once the expression inside the parentheses was simplified, we moved on to multiply and then divide.
• Finally, handle addition and subtraction, also from left to right. Again, in this example, no addition or subtraction remained after addressing the parentheses and multiplication.

By following these steps, you'll ensure that expressions are simplified consistently and correctly. Always remember to go from left to right for operations of the same precedence.

Multiplication

Multiplication is one of the four basic arithmetic operations and is often considered repeated addition. In our exercise, the multiplication was done after simplifying the expression inside the parentheses, and before performing the division.
To multiply fractions and whole numbers, the rule is simple: multiply the numerators by each other and the denominators by each other. However, in our example, we have something slightly different as we multiplied a fraction by a whole number.

• To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction and keep the denominator the same.
• For the expression \( \frac{5}{9} \times 45 \), you calculate \( 5 \times 45 \) first, which equals 225.
• Write this as \( \frac{225}{9} \), preparing it for the next operation—division.

Understanding this operation ensures you can transition from multiplication to the next step without errors.

Division

Division is the arithmetic operation used to determine how many times one number is contained within another. It is essentially the opposite of multiplication and is also one of the key components to simplifying expressions.
In the final step of our exercise, once we obtained \( \frac{225}{9} \) from the multiplication process, division was necessary to reach the simplified result.

• Division can be understood as determining how many groups of the divisor 'fit' into the dividend. In this exercise, we divided 225 by 9.
• The calculation \( 225 \div 9 \) was performed to find the quotient, which turned out to be 25.
• This is the final answer when simplifying the original expression, demonstrating that the expression \( \frac{5}{9}(77-32) \) simplifies neatly to the number 25.

Division requires careful attention as it is the last step in the order of operations sequence in our example, ensuring the problem is solved completely and accurately.