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Operations on Large Numbers - Word Problems on Four Operations

Grade 5ICSE

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

🔑Concepts

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Addition of Large Numbers: This operation is used when word problems use terms like 'total', 'sum', 'altogether', or 'increased by'. To solve these, align the numbers in a vertical place value chart starting from the Ones (OO) on the right up to Crores (CC) or Ten-Crores (TCTC) on the left. Imagine stacking blocks of different sizes where each column must be added carefully, carrying over any value ten or greater to the next column on the left.

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Subtraction of Large Numbers: Subtraction is indicated by keywords such as 'difference', 'remaining', 'how much more', or 'left'. Visually, this is like removing a smaller portion from a larger whole. When subtracting large numbers with many zeros (like 1,00,00,0001,00,00,000), you must 'borrow' or regroup from the nearest non-zero place value, which visually looks like a chain reaction moving from left to right across the place value columns.

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Multiplication of Large Numbers: Multiplication is used for 'scaling up' or repeated addition, often when you know the value of one item and need to find the total value of many. For example, finding the total weight of 5,0005,000 identical crates. Think of this as a large grid or an area model where the number of rows and columns represents the two large numbers being multiplied.

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Division of Large Numbers: This operation is used to 'split' or distribute a large quantity into equal smaller groups, or to find the cost or value of a single unit. Keywords include 'each', 'share equally', and 'average'. Visually, imagine a large pile being partitioned into several equal smaller piles. The 'Remainder' represents the piece that is left over because it is too small to form a full equal group.

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Identifying Operations in Context: Success in word problems depends on translating words into math. If a problem asks for a total after a series of increases, you visualize the number getting bigger (Addition). If it asks for the 'gap' between two large quantities like the populations of two countries, you visualize the difference (Subtraction).

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The Unitary Method: This is a visual two-step process. First, you divide to find the value of 'one' unit (e.g., the price of 1 pen). Then, you multiply that single value to find the total for the required number of units. It’s like finding the size of one building block before calculating the size of a whole tower.

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Estimation and Reasonable Answers: For very large numbers, it is helpful to round the numbers to the nearest Lakh or Crore before calculating. This provides a mental 'picture' of what the answer should look like. If your exact sum or product is significantly different from your estimate, it usually means a digit was misplaced in the place value columns.

📐Formulae

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

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

Product=Multiplicand×Multiplier\text{Product} = \text{Multiplicand} \times \text{Multiplier}

Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}

Value of one unit=Total ValueTotal Number of units\text{Value of one unit} = \frac{\text{Total Value}}{\text{Total Number of units}}

Average=Sum of all quantitiesNumber of quantities\text{Average} = \frac{\text{Sum of all quantities}}{\text{Number of quantities}}

💡Examples

Problem 1:

A glass factory produced 3,45,67,8903,45,67,890 bottles in the year 2022 and 4,12,34,5604,12,34,560 bottles in the year 2023. How many bottles were produced in total over the two years?

Solution:

  1. Bottles in 2022 = 3,45,67,8903,45,67,890 \ 2. Bottles in 2023 = 4,12,34,5604,12,34,560 \ 3. Total production = 3,45,67,890+4,12,34,5603,45,67,890 + 4,12,34,560 \ 4. Align by place value: \ 3,45,67,8903,45,67,890 \ +4,12,34,560+ 4,12,34,560 \ −−−−−−−−−−−−------------ \ 7,58,02,4507,58,02,450

Explanation:

The word 'total' indicates we need to add the two quantities. We arrange the numbers in columns according to the Indian Place Value system (Crores, Lakhs, Thousands, Ones) and add from right to left, carrying over values where necessary.

Problem 2:

A charity wants to distribute 1,25,00,0001,25,00,000 kilograms of rice equally among 500500 villages. How many kilograms of rice will each village receive?

Solution:

  1. Total rice = 1,25,00,0001,25,00,000 kg \ 2. Total villages = 500500 \ 3. Rice per village = 1,25,00,000÷5001,25,00,000 \div 500 \ 4. Simplify by cancelling zeros: 1,25,00000500\frac{1,25,000 \cancel{00}}{5 \cancel{00}} \ 5. Perform the division: 1,25,000÷5=25,0001,25,000 \div 5 = 25,000

Explanation:

The phrase 'distribute equally' tells us to use division. We divide the total quantity of rice by the number of villages. By cancelling two zeros from both the dividend and the divisor, the calculation becomes much simpler, resulting in 25,00025,000 kg per village.