In a corporate training session on financial literacy, employees are taught the mnemonic 'Please Excuse My Dear Aunt Sally' to remember the order of operations. What does the word 'Dear' represent in this acrostic?

The sentence "Please Excuse My Dear Aunt Sally" is a widely used acrostic in mathematics that helps students remember the correct order of operations: **P**arentheses, **E**xponents, **M**ultiplication, **D**ivision, **A**ddition, and **S**ubtraction. An important nuance is that "My Dear" (multiplication and division) share equal priority and are performed left to right, and "Aunt Sally" (addition and subtraction) also share equal priority and are performed left to right—so the mnemonic does not mean multiplication always comes before division or addition always before subtraction.

Google

Claude

An acrostic is a mnemonic strategy that involves creating a memorable phrase or sentence in which the first letter of each word corresponds to an item or concept that needs to be remembered in a particular sequence.

Acrostic as a Mnemonic Device

The **order of operations** is a universally agreed-upon set of rules that dictates the sequence in which operations must be performed when simplifying a mathematical expression, ensuring that everyone arrives at the same result. The four steps, applied in order, are:

1. **Parentheses and Grouping Symbols** — Simplify all expressions inside parentheses, brackets, or braces first, beginning with the innermost group.
2. **Exponents** — Evaluate all expressions involving exponents.
3. **Multiplication and Division** — Perform all multiplication and division from left to right. These two operations share equal priority; neither is automatically performed before the other.
4. **Addition and Subtraction** — Perform all addition and subtraction from left to right. These two operations also share equal priority.

Without these conventions, the expression $$4 + 3 \cdot 7$$ could reasonably be interpreted as either $$25$$ or $$49$$, depending on whether multiplication or addition is performed first.

Order of Operations in Algebra

https://openstax.org/details/books/elementary-algebra-2e

OpenStax Elementary Algebra

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

Example of an Acrostic: 'Please Excuse My Dear Aunt Sally'

To simplify an expression that combines integer multiplication with addition and subtraction, follow the order of operations — perform all multiplications before any additions or subtractions:

1. **Multiply first:** Compute each product: $$7(-2) = -14$$ (different signs produce a negative) and $$4(-7) = -28$$ (different signs produce a negative). The expression becomes $$-14 + (-28) - 6$$.
2. **Add:** Combine $$-14 + (-28) = -42$$ (same signs — add the absolute values and keep the negative sign). The expression becomes $$-42 - 6$$.
3. **Subtract:** Compute $$-42 - 6 = -48$$.

The final result is $$-48$$. This problem shows that when an expression mixes multiplication with addition and subtraction and contains no grouping symbols, all multiplications must be completed first; then the additions and subtractions are performed from left to right.

Simplifying $$7(-2) + 4(-7) - 6$$

When an expression contains a parenthesized sub-expression that is multiplied by a factor, the order of operations requires simplifying inside the parentheses before performing the multiplication and any remaining operations:

1. **Parentheses first:** Compute $$9 - 12 = -3$$ inside the parentheses, giving $$12 - 3(-3)$$.
2. **Multiply:** Compute $$3(-3) = -9$$ (different signs yield a negative product), giving $$12 - (-9)$$.
3. **Subtract the negative:** Subtracting $$-9$$ is equivalent to adding $$9$$: $$12 + 9 = 21$$.

The final result is $$21$$. A common error is to multiply $$3 \times 9$$ or $$3 \times 12$$ before resolving the subtraction inside the parentheses. The parentheses override normal left-to-right processing by requiring their contents to be fully evaluated first.

Simplifying $$12 - 3(9 - 12)$$

When an expression contains multiple subtractions with parentheses, apply the order of operations alongside integer subtraction rules:

1. **Parentheses first:** Compute $$-4 - 3 = -7$$ inside the parentheses, giving $$7 - (-7) - 9$$.
2. **Subtract the negative:** Since subtracting a negative is equivalent to adding a positive, $$7 - (-7) = 7 + 7 = 14$$, so the expression becomes $$14 - 9$$.
3. **Subtract left to right:** Compute $$14 - 9 = 5$$.

The final result is $$5$$. This problem illustrates how two skills work together: the order of operations determines that the parenthesized expression $$(-4 - 3)$$ must be resolved first, and the rule $$a - (-b) = a + b$$ converts the subtraction of the negative result into addition. The remaining operations are then carried out from left to right.

Simplifying $$7 - (-4 - 3) - 9$$

To simplify an expression that combines exponents, multiplication, and division with integers, apply the order of operations — evaluate exponents before performing multiplication or division:

1. **Exponents first:** Evaluate $$(-2)^3 = (-2)(-2)(-2) = -8$$. The base is negative and the exponent is odd, so the result is negative. The expression becomes $$8(-9) \div (-8)$$.
2. **Multiply:** Compute $$8(-9) = -72$$ (different signs produce a negative product). The expression becomes $$-72 \div (-8)$$.
3. **Divide:** Compute $$-72 \div (-8) = 9$$ (same signs produce a positive quotient).

The final result is $$9$$. This problem demonstrates how exponents, multiplication, and division interact in a single expression: exponents are always resolved first, then multiplication and division are performed from left to right.

Simplifying $$8(-9) \div (-2)^3$$

When an expression mixes division, multiplication, and addition without grouping symbols, the order of operations requires completing all multiplications and divisions (from left to right) before any additions:

1. **Divide first (left to right):** Because division appears to the left of the multiplication, compute $$-30 \div 2 = -15$$ first. The expression becomes $$-15 + (-3)(-7)$$.
2. **Multiply:** Compute $$(-3)(-7) = 21$$ (same signs produce a positive product). The expression becomes $$-15 + 21$$.
3. **Add:** Compute $$-15 + 21 = 6$$ (different signs — subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value).

The final result is $$6$$. This example highlights that multiplication and division share equal priority, so when both appear in an expression they are performed left to right — not multiplication before division. Here the division $$-30 \div 2$$ is evaluated first simply because it appears further to the left.

Simplifying $$-30 \div 2 + (-3)(-7)$$

To simplify a complex fraction whose numerator or denominator contains an expression (not just a single fraction), follow these three steps:

1. **Simplify the numerator.** Apply the order of operations to reduce the numerator to a single fraction or number.
2. **Simplify the denominator.** Apply the order of operations to reduce the denominator to a single fraction or number.
3. **Divide the numerator by the denominator.** Rewrite the main fraction bar as a division problem, then multiply by the reciprocal and simplify if possible.

Because the numerator and denominator may involve addition, subtraction, exponents, or other operations, each part must be fully evaluated on its own before the final division step is performed.

Simplifying Complex Fractions

When an expression is written as a fraction and the numerator or denominator (or both) contain operations — such as addition, subtraction, multiplication, or exponents — the fraction bar functions as a grouping symbol. It signals that the expressions on top and on bottom must each be fully simplified before the overall fraction is evaluated. Follow a two-step process:

1. **Simplify the numerator and the denominator separately.** Apply the order of operations to reduce each part to a single number or simplest form.
2. **Simplify the resulting fraction.** Once the numerator and denominator are each a single value, divide and reduce the fraction to lowest terms if possible.

For example, to simplify $$\frac{5 + 3}{10 - 2}$$: first compute the numerator $$5 + 3 = 8$$ and the denominator $$10 - 2 = 8$$, then simplify $$\frac{8}{8} = 1$$. The key insight is that the fraction bar groups the entire numerator expression and the entire denominator expression, just as parentheses would — all operations above and below the bar must be completed before dividing.

Simplifying an Expression with a Fraction Bar

To simplify the expression $$18 \div 6 + 4(5 - 2)$$, apply the order of operations step-by-step.

**Step 1. Parentheses:** Evaluate the expression inside the parentheses first. Subtract $$5 - 2 = 3$$, making the expression $$18 \div 6 + 4(3)$$.

**Step 2. Exponents:** There are no exponents, so proceed to multiplication and division.

**Step 3. Multiplication and Division:** Perform these operations from left to right. First, divide $$18 \div 6 = 3$$, resulting in $$3 + 4(3)$$. Next, multiply $$4 \cdot 3 = 12$$, which gives $$3 + 12$$.

**Step 4. Addition and Subtraction:** Finally, perform the remaining addition to find the result: $$3 + 12 = 15$$.

Simplifying $$18 \div 6 + 4(5 - 2)$$

These two expressions demonstrate how grouping symbols alter the result when the same numbers and operations are involved:

**Without parentheses** — $$4 + 3 \cdot 7$$: There are no parentheses or exponents, so move to multiplication and division. Compute $$3 \cdot 7 = 21$$ first, then add: $$4 + 21 = 25$$.

**With parentheses** — $$(4 + 3) \cdot 7$$: Parentheses take highest priority, so compute $$4 + 3 = 7$$ first, then multiply: $$7 \cdot 7 = 49$$.

The identical numbers and operations yield different results ($$25$$ versus $$49$$) solely because the parentheses force the addition to be evaluated before the multiplication.

Simplifying $$4 + 3 \cdot 7$$ and $$(4 + 3) \cdot 7$$

To simplify an expression with nested grouping symbols like $$5 + 2^3 + 3[6 - 3(4 - 2)]$$, start from the innermost group and work outward following the order of operations:

1. **Innermost Parentheses**: Subtract inside the parentheses $$(4 - 2) = 2$$, yielding $$5 + 2^3 + 3[6 - 3(2)]$$.
2. **Inside Brackets (Multiplication)**: Multiply $$3(2) = 6$$, resulting in $$5 + 2^3 + 3[6 - 6]$$.
3. **Inside Brackets (Subtraction)**: Subtract $$6 - 6 = 0$$, simplifying the brackets to $$5 + 2^3 + 3[0]$$.
4. **Exponents**: Evaluate the power $$2^3 = 8$$, giving $$5 + 8 + 3[0]$$.
5. **Multiplication**: Multiply $$3[0] = 0$$, yielding $$5 + 8 + 0$$.
6. **Addition**: Add the remaining numbers to get the final answer: $$13$$.

Simplifying $$5 + 2^3 + 3[6 - 3(4 - 2)]$$

To simplify the expression $$9 + 5^3 - [4(9 + 3)]$$, apply the order of operations by resolving the innermost grouping symbols first:

1. **Innermost Parentheses**: Add inside the parentheses $$(9 + 3) = 12$$, giving $$9 + 5^3 - [4(12)]$$.
2. **Inside Brackets**: Multiply $$4(12) = 48$$, resulting in $$9 + 5^3 - [48]$$, which simplifies to $$9 + 5^3 - 48$$.
3. **Exponents**: Evaluate the exponent $$5^3 = 125$$, yielding $$9 + 125 - 48$$.
4. **Addition and Subtraction**: Working from left to right, first add $$9 + 125 = 134$$, then subtract $$134 - 48 = 86$$.

The final result is $$86$$.

Simplifying $$9 + 5^3 - [4(9 + 3)]$$

To simplify the expression $$7^2 - 2[4(5 + 1)]$$, apply the order of operations, starting with the most deeply nested grouping symbols:

1. **Innermost Parentheses**: Add inside the parentheses $$(5 + 1) = 6$$, resulting in $$7^2 - 2[4(6)]$$.
2. **Inside Brackets**: Multiply $$4(6) = 24$$, giving $$7^2 - 2[24]$$.
3. **Exponents**: Evaluate the power $$7^2 = 49$$, yielding $$49 - 2[24]$$.
4. **Multiplication**: Multiply $$2[24] = 48$$, which gives $$49 - 48$$.
5. **Subtraction**: Subtract the final values to obtain the answer: $$1$$.

Simplifying $$7^2 - 2[4(5 + 1)]$$

To simplify the expression $$30 \div 5 + 10(3 - 2)$$, we apply the order of operations. First, evaluate the expression inside the parentheses by subtracting: $$3 - 2 = 1$$. The expression becomes $$30 \div 5 + 10(1)$$. Next, perform multiplication and division from left to right. Divide $$30$$ by $$5$$ to get $$6$$, resulting in $$6 + 10(1)$$. Then, multiply $$10$$ by $$1$$ to get $$10$$, which gives $$6 + 10$$. Finally, add $$6$$ and $$10$$ to get the simplified result, $$16$$.

Simplifying $$30 \div 5 + 10(3 - 2)$$

To simplify the expression $$70 \div 10 + 4(6 - 2)$$, we follow the order of operations. First, perform the operation inside the parentheses: $$6 - 2 = 4$$. The expression updates to $$70 \div 10 + 4(4)$$. Next, execute multiplication and division from left to right. Dividing $$70$$ by $$10$$ gives $$7$$, so the expression becomes $$7 + 4(4)$$. Multiplying $$4$$ by $$4$$ yields $$16$$, resulting in $$7 + 16$$. Finally, add $$7$$ and $$16$$ to reach the fully simplified value of $$23$$.

Learn Before

Related

Learn After