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Number Sense and Operations - Addition and Subtraction of Multi-digit Numbers

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Place Value Alignment: When adding or subtracting multi-digit numbers, digits must be aligned vertically according to their place value (ones under ones, tens under tens, etc.). Visualize a vertical grid where the decimal points or the right-most digits are perfectly stacked to ensure you are adding the same units.

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Regrouping in Addition (Carrying): If the sum of digits in a column is 1010 or greater, you must regroup. For example, in the ones column, if the sum is 1313, you write the 33 in the ones place and carry the 11 (representing 1010) to the top of the tens column. This is visually represented by a small digit placed above the next column to the left.

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Regrouping in Subtraction (Borrowing): If the top digit (minuend) is smaller than the bottom digit (subtrahend), you must 'borrow' from the next place value to the left. Visualize crossing out a digit in the tens place, decreasing it by 11, and adding 1010 to the digit in the ones place to make the subtraction possible.

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Subtracting Across Zeros: When borrowing from a column that contains a 00, you must move further left to the first non-zero digit. For a number like 5,0005,000, you borrow from the thousands place (55 becomes 44), making the hundreds 99, the tens 99, and the ones 1010. Visually, this creates a chain of crossed-out zeros replaced by nines.

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Estimation for Reasonableness: Before calculating the exact answer, round the numbers to the nearest thousand or ten-thousand to estimate the result. If your exact answer is significantly different from your estimate, you should check your regrouping steps.

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Properties of Addition: The Commutative Property (a+b=b+aa + b = b + a) means the order of numbers doesn't change the sum. The Associative Property ((a+b)+c=a+(b+c)(a + b) + c = a + (b + c)) means the grouping of numbers doesn't change the sum. These properties help in mental math and checking work.

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Inverse Operations: Addition and subtraction are inverse operations. You can check the result of a subtraction problem (Minuend−Subtrahend=DifferenceMinuend - Subtrahend = Difference) by adding the difference and the subtrahend together (Difference+Subtrahend=MinuendDifference + Subtrahend = Minuend).

📐Formulae

Addend1+Addend2=Sum\text{Addend}_1 + \text{Addend}_2 = \text{Sum}

Minuend−Subtrahend=Difference\text{Minuend} - \text{Subtrahend} = \text{Difference}

a+b=b+a (Commutative Property)a + b = b + a \text{ (Commutative Property)}

(a+b)+c=a+(b+c) (Associative Property)(a + b) + c = a + (b + c) \text{ (Associative Property)}

Difference+Subtrahend=Minuend (Check)\text{Difference} + \text{Subtrahend} = \text{Minuend} \text{ (Check)}

💡Examples

Problem 1:

Find the sum of 45,78245,782 and 36,41936,419.

Solution:

45,782+36,419=82,20145,782 + 36,419 = 82,201

  1. Align columns: 2+9=112 + 9 = 11 (write 11, carry 11) 8+1+1(carry)=108 + 1 + 1 (carry) = 10 (write 00, carry 11) 7+4+1(carry)=127 + 4 + 1 (carry) = 12 (write 22, carry 11) 5+6+1(carry)=125 + 6 + 1 (carry) = 12 (write 22, carry 11) 4+3+1(carry)=84 + 3 + 1 (carry) = 8 Result: 82,20182,201

Explanation:

This problem requires regrouping in almost every column. Start from the ones place and move left, always remembering to add the 'carried' digit from the previous column.

Problem 2:

Subtract 24,56724,567 from 60,00060,000.

Solution:

60,000−24,567=35,43360,000 - 24,567 = 35,433

  1. Borrow from the 66 in the ten-thousands place.
  2. The 66 becomes 55.
  3. The zeros in the thousands, hundreds, and tens places become 99s.
  4. The zero in the ones place becomes 1010.
  5. Subtract: 10−7=310 - 7 = 3 (ones) 9−6=39 - 6 = 3 (tens) 9−5=49 - 5 = 4 (hundreds) 9−4=59 - 4 = 5 (thousands) 5−2=35 - 2 = 3 (ten-thousands) Result: 35,43335,433

Explanation:

This example demonstrates borrowing across multiple zeros. Since you cannot subtract 77 from 00, you must go to the first non-zero digit (the 66) to regroup across all place values.