(50 + 2) + 5-5
Find the differende
256,1805
136.667
a
Find the quotient.
Find the quoient.
93.784
43,07 x 1.8
Find the product
23,70
Use the Distributive ProDerty to express14 63
Step-by-Step Solutions for the Worksheet Problems
Here is the complete, step-by-step breakdown for each of the math problems presented on the worksheet.
1. Order of Operations: (50 + 2) + 5^3 - 5
RULE / FORMULA USED: PEMDAS / Order of Operations
- Parentheses: Solve operations inside grouping symbols first.
- Exponents: Evaluate powers.
- Multiplication and Division: Left to right.
- Addition and Subtraction: Left to right.
SOLUTION STEPS:
Step 1 — Evaluate the expression inside parentheses
(50 + 2) = 52
Step 2 — Evaluate the exponent
5^3 = 5 \times 5 \times 5 = 125
Step 3 — Substitute the values back into the expression
52 + 125 - 5
Step 4 — Perform addition and subtraction from left to right
52 + 125 = 177
177 - 5 = 172
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ANSWER: 172
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2. Find the Difference: 256.805 - 136.667
RULE / FORMULA USED: Line up the decimals vertically and subtract column by column from right to left, borrowing when necessary.
SOLUTION STEPS:
Step 1 — Set up the vertical subtraction
$$\begin{array}{r@{\quad}l}
256.805 \
- 136.667 \
\hline
\end{array}$$
Step 2 — Subtract the thousandths place
- Borrow 1 from the tenths place (8 becomes 7, hundredths becomes 9, thousandths becomes 15):
15 - 7 = 8
Step 3 — Subtract the hundredths and tenths places
- Hundredths: 9 - 6 = 3
- Tenths: 7 - 6 = 1
Step 4 — Subtract the whole number parts
- Ones place: 6 - 6 = 0
- Tens place: 5 - 3 = 2
- Hundreds place: 2 - 1 = 1
$$\begin{array}{r@{\quad}l}
256.805 \
- 136.667 \
\hline
120.138
\end{array}$$
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ANSWER: 120.138
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3. Find the Quotient: \frac{8}{11} \div \frac{3}{5}
RULE / FORMULA USED: Fraction Division (Keep, Change, Flip)
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
SOLUTION STEPS:
Step 1 — Multiply by the reciprocal of the divisor
\frac{8}{11} \div \frac{3}{5} = \frac{8}{11} \times \frac{5}{3}
Step 2 — Multiply numerators and denominators
\text{Numerator: } 8 \times 5 = 40
\text{Denominator: } 11 \times 3 = 33
\frac{40}{33}
Step 3 — Convert to a mixed number
40 \div 33 = 1 \text{ with a remainder of } 7 \implies 1\frac{7}{33}
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ANSWER: \frac{40}{33} or 1\frac{7}{33}
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4. Find the Quotient: 3,784 \div 18
RULE / FORMULA USED: Long Division Algorithm (Divide, Multiply, Subtract, Bring Down).
SOLUTION STEPS:
Step 1 — Divide into the first two digits
- 18 goes into 37 two times (2 \times 18 = 36).
- Subtract: 37 - 36 = 1.
Step 2 — Bring down the next digit (8)
- The new number is 18.
- 18 goes into 18 one time (1 \times 18 = 18).
- Subtract: 18 - 18 = 0.
Step 3 — Bring down the last digit (4)
- The new number is 4.
- 18 goes into 4 zero times (0 \times 18 = 0).
- Subtract: 4 - 0 = 4 (Remainder).
Step 4 — State the final quotient
3,784 \div 18 = 210 \text{ R } 4 = 210\frac{4}{18} = 210\frac{2}{9} \approx 210.2\bar{2}
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ANSWER: 210 \text{ R } 4 (or 210\frac{2}{9} / 210.2\bar{2})
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5. Find the Product: 43.07 \times 1.8
RULE / FORMULA USED: Multiply the numbers as whole numbers, then place the decimal point so that the product has the total number of decimal places from both factors (2 + 1 = 3 decimal places).
SOLUTION STEPS:
Step 1 — Multiply without decimals
$$\begin{array}{r@{\quad}l}
4307 \
\times \quad 18 \
\hline
34456 & (4307 \times 8) \
- 43070 & (4307 \times 10) \
\hline
77526
\end{array}$$
Step 2 — Count and place the decimal places
- 43.07 has 2 decimal places.
- 1.8 has 1 decimal place.
- Total decimal places needed = 2 + 1 = 3.
77526 \rightarrow 77.526
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ANSWER: 77.526
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6. Find the Quotient: 23.70 \div 1.8
RULE / FORMULA USED: Make the divisor a whole number by shifting the decimal point one place to the right in both the divisor and the dividend:
23.70 \div 1.8 = 237.0 \div 18 = \frac{237}{18}
SOLUTION STEPS:
Step 1 — Simplify the fraction by dividing numerator and denominator by 3
\frac{237 \div 3}{18 \div 3} = \frac{79}{6}
Step 2 — Perform long division
79 \div 6 = 13 \text{ with a remainder of } 1
\frac{79}{6} = 13\frac{1}{6} = 13.1666...
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ANSWER: 13.1\bar{6} (or 13\frac{1}{6})
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7. Distributive Property: Express 14 + 63
RULE / FORMULA USED: Factoring out the Greatest Common Factor (GCF) using the Distributive Property:
a \cdot b + a \cdot c = a(b + c)
SOLUTION STEPS:
Step 1 — Find the prime factors of each term
- Factors of 14: 1, 2, \mathbf{7}, 14
- Factors of 63: 1, 3, \mathbf{7}, 9, 21, 63
- \text{GCF}(14, 63) = 7
Step 2 — Rewrite each term as a multiple of the GCF
14 = 7 \times 2
63 = 7 \times 9
Step 3 — Factor out the GCF using the Distributive Property
14 + 63 = 7(2 + 9)
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ANSWER: 7(2 + 9)
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KEY CONCEPTS SUMMARY:
- Order of Operations: Exponents and parentheses always take priority over basic addition/subtraction.
- Fraction Division: Invert the second fraction and multiply.
- Distributive Property Factoring: Find the GCF of the numbers and factor it out in front of parentheses.
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Solutions for the Worksheet Problems
RULE / FORMULA USED:
- Order of Operations: Evaluate parentheses, exponents, multiplication/division, then addition/subtraction.
- Decimal Subtraction & Multiplication: Align decimal places for subtraction; sum decimal places for multiplication.
- Fraction Division: \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
- Distributive Property: a \cdot b + a \cdot c = a(b + c)
SOLUTION STEPS:
1. (50 + 2) + 5^3 - 5
Step 1 — Evaluate parentheses and exponent
- (50 + 2) = 52
- 5^3 = 125
Step 2 — Add and subtract from left to right
52 + 125 - 5 = 177 - 5 = 172
2. 256.805 - 136.667
Step 1 — Subtract column by column with regrouping
256.805 - 136.667 = 120.138
3. \frac{8}{11} \div \frac{3}{5}
Step 1 — Multiply by the reciprocal
\frac{8}{11} \times \frac{5}{3} = \frac{8 \times 5}{11 \times 3} = \frac{40}{33} = 1\frac{7}{33}
4. 3,784 \div 18
Step 1 — Divide with remainder
3,784 = 18 \times 210 + 4
\text{Quotient} = 210 \text{ R } 4 = 210\frac{2}{9} \approx 210.2\bar{2}
5. 43.07 \times 1.8
Step 1 — Multiply integers and place 3 decimal digits
4307 \times 18 = 77526 \implies 77.526
6. 23.70 \div 1.8
Step 1 — Shift decimal point one place and divide
\frac{237}{18} = \frac{79}{6} = 13\frac{1}{6} \approx 13.1\bar{6}
7. Distributive Property for 14 + 63
Step 1 — Factor out the Greatest Common Factor (\text{GCF} = 7)
14 + 63 = 7(2 + 9)
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ANSWERS:
- 172
- 120.138
- \frac{40}{33} (or 1\frac{7}{33})
- 210 \text{ R } 4 (or 210\frac{2}{9})
- 77.526
- 13.1\bar{6} (or 13\frac{1}{6})
- 7(2 + 9)
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KEY CONCEPTS:
1. Greatest Common Factor (GCF)
- Definition: The highest number that divides evenly into two or more numbers.
- In this problem: Factoring out 7 from both 14 and 63 allows rewriting the sum as a product: 7(2 + 9).
COMMON MISTAKES:
Skipping the Reciprocal in Fraction Division
- Wrong: \frac{8}{11} \div \frac{3}{5} = \frac{8 \div 3}{11 \div 5}
- Right: \frac{8}{11} \times \frac{5}{3} = \frac{40}{33}
- Why it’s wrong: Dividing fractions requires multiplying by the inverted second fraction (reciprocal).
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