RBSE Solutions Class 9 Maths Chapter 1 Vedic Mathematics Exercise 1.1

By Sutra Ekadhikena Purvena, Add the Following:

 

Question 1.

Answer: The sum is calculated by vertically adding each column, starting from the rightmost digit. When a column's sum exceeds 9, the 'Ekadhikena Purvena' principle involves placing a dot on the previous digit in the next row to indicate a carry-over, simplifying the addition process. This method helps manage carries in a structured way.In simple words: We add the numbers column by column from right to left. When the sum in a column is more than nine, we put a dot on the next number to the left in the row below it, and carry over the extra part. The final total is 312047.

🎯 Exam Tip: When using the Ekadhikena Purvena method for addition, remember to place the dot on the 'Ekadhika' (one more) digit of the next number in the following column whenever a sum exceeds a base value (usually 10).

 

Question 2.
89789
97686
76978
86798
Answer: The numbers are added vertically, column by column, from right to left. Any value carried over from one column is added to the next column on the left.

The total sum obtained is 351251. This is a straightforward process of carrying tens to the next place value.In simple words: Add the numbers in each column, starting from the very right. If a sum is 10 or more, carry the 'tens' digit to the next column to the left. The final answer is 351251.

🎯 Exam Tip: Always align numbers by their place value (units, tens, hundreds, etc.) before performing vertical addition to avoid errors.

 

Question 3.
kg gm
178 45
246 725
569 188
45 894
Answer: We add the 'gm' (grams) and 'kg' (kilograms) parts separately. When the sum of grams exceeds 1000 gm, we carry over 1 kg for every 1000 gm to the kilogram column, since 1 kg = 1000 gm. The dot notation indicates a carry-over to the next place value.

Sum of grams: \( 45 + 725 + 188 + 894 = 1852 \) gm. Since \( 1852 \) gm \( = 1 \) kg \( 852 \) gm, we carry \( 1 \) kg to the kilogram column. Sum of kilograms: \( 178 + 246 + 569 + 45 + 1 \) (carry-over) \( = 1039 \) kg. The final total is 1039 kg and 852 gm.In simple words: First, add all the grams. If the total grams are more than 1000, take out 1000 grams and add 1 kilogram to the kilogram sum. Then add all the kilograms, including any extra kilogram from the grams. The final result is 1039 kg and 852 gm.

🎯 Exam Tip: When adding quantities with units like kg/gm or km/m/cm, always remember the conversion factors (e.g., 1 kg = 1000 gm, 1 km = 1000 m, 1 m = 100 cm) to handle carry-overs correctly between units.

 

Question 4.
km m cm
25 510 36
47 85 52
18 123 75
53 805 28
Answer: We add the 'cm' (centimeters), 'm' (meters), and 'km' (kilometers) parts separately. We use the conversion factors: 1 m = 100 cm and 1 km = 1000 m, to carry over values to the next higher unit.

Sum of cm: \( 36 + 52 + 75 + 28 = 191 \) cm. Since \( 191 \) cm \( = 1 \) m \( 91 \) cm, we carry \( 1 \) m to the meters column. Sum of m: \( 510 + 85 + 123 + 805 + 1 \) (carry-over) \( = 1524 \) m. Since \( 1524 \) m \( = 1 \) km \( 524 \) m, we carry \( 1 \) km to the kilometers column. Sum of km: \( 25 + 47 + 18 + 53 + 1 \) (carry-over) \( = 144 \) km. The final total is 144 km, 524 m, and 91 cm.In simple words: Add centimeters first, then meters, then kilometers. For centimeters, every 100 cm makes 1 meter, so carry it over. For meters, every 1000 meters makes 1 kilometer, so carry it over. The total is 144 km, 524 m, and 91 cm.

🎯 Exam Tip: Always pay close attention to the base of each unit (e.g., 100 for cm to m, 1000 for m to km) to ensure correct carry-overs during addition.

By Vedic Method, Subtract the Following:

 

Question 5.
746
- 389
Answer: We use the Vedic subtraction method, which often involves using complements or 'all from 9 and the last from 10' for digits that cannot be subtracted directly. Here, we can think of it as standard subtraction with borrowing.

Starting from the right: 1. \( 6 - 9 \): We cannot subtract 9 from 6. We borrow from the tens place (4 becomes 3), making 6 into 16. \( 16 - 9 = 7 \). 2. \( 3 - 8 \): We cannot subtract 8 from 3. We borrow from the hundreds place (7 becomes 6), making 3 into 13. \( 13 - 8 = 5 \). 3. \( 6 - 3 = 3 \). The final difference is 357. This method makes calculations quicker by focusing on compliments.In simple words: We subtract numbers from right to left. If the bottom number is bigger than the top number, we borrow from the digit to its left. Then we subtract. The answer is 357.

🎯 Exam Tip: Vedic subtraction often uses the 'All from 9 and the last from 10' rule, converting subtractions to additions with complements, which can be faster for certain number combinations.

 

Question 6.
4032
- 3543
Answer: We subtract the numbers using the Vedic method, which is similar to standard borrowing. Each digit is subtracted from right to left, handling any necessary borrowing from the adjacent higher place value.

1. \( 2 - 3 \): Borrow from 3 (becomes 2), so 2 becomes 12. \( 12 - 3 = 9 \). 2. \( 2 - 4 \): Borrow from 0 (which itself borrows from 4, making 4 into 3, 0 into 9), so 2 becomes 12. \( 12 - 4 = 8 \). 3. \( 9 - 5 = 4 \). 4. \( 3 - 3 = 0 \). The difference is 489. This method reduces the chances of errors when dealing with multiple borrowings.In simple words: We subtract from right to left. If a digit on top is smaller, we "borrow" 1 from the number to its left, which makes the top digit 10 bigger. Then we subtract. The answer is 489.

🎯 Exam Tip: Practice with numbers involving zeros in the minuend (the top number) to master borrowing across multiple place values efficiently.

 

Question 8.
8317
- 6454
Answer: This subtraction uses the Vedic method with dot notation, indicating a complement or borrow from the previous digit. We subtract each column from right to left, adjusting for borrowing.

1. \( 7 - 4 = 3 \). 2. \( 1 - 5 \): Borrow from 3 (becomes 2), so 1 becomes 11. \( 11 - 5 = 6 \). 3. \( 2 - 4 \): Borrow from 8 (becomes 7), so 2 becomes 12. \( 12 - 4 = 8 \). 4. \( 7 - 6 = 1 \). The final difference is 1863. This structured approach helps in maintaining accuracy across digits.In simple words: We subtract the bottom number from the top number, starting from the right. When a top digit is smaller than the bottom digit, we borrow from the digit next to it on the left. The final difference is 1863.

🎯 Exam Tip: Using the 'dot' or 'Ekadhikena' system for borrowing means adding one to the digit being borrowed from in the subtrahend (bottom number) and using the complement of the minuend (top number), which can make subtraction faster.

By Sutra Ekadhikena Purvena, Multiply the Following:

 

Question 9. 42 x 48
Answer: The Ekadhikena Purvena sutra is used here for multiplying numbers whose last digits add up to 10 (or 100, etc.) and whose preceding digits are the same. The numbers are 42 and 48. The first digits are the same: 4. The last digits sum to 10: \( 2 + 8 = 10 \). So, we can apply the sutra. 1. Multiply the common first digit by 'one more than itself': \( 4 \times (4+1) = 4 \times 5 = 20 \). This forms the left part of the answer. 2. Multiply the last digits: \( 2 \times 8 = 16 \). This forms the right part of the answer. 3. Combine the parts: 20 and 16. So, \( 42 \times 48 = 2016 \). This method simplifies multiplication for specific types of numbers.In simple words: For numbers like 42 and 48, where the first digit is the same (4) and the last digits add up to 10 (2+8=10), we multiply the first digit (4) by one more than itself (5), which is 20. Then we multiply the last digits (2 and 8), which is 16. Put these two results together: 2016.

🎯 Exam Tip: This sutra is very effective for numbers like \( 31 \times 39 \), \( 63 \times 67 \), or \( 104 \times 106 \), where the tens (or hundreds) digit is the same and the unit digits sum to 10.

 

Question 10. 103 x 107
Answer: We use the Ekadhikena Purvena sutra for \( 103 \times 107 \). The first digits (excluding the units) are the same: 10. The last digits (units place) sum to 10: \( 3 + 7 = 10 \). 1. Multiply the common first digit by 'one more than itself': \( 10 \times (10+1) = 10 \times 11 = 110 \). This is the left part. 2. Multiply the last digits: \( 3 \times 7 = 21 \). This is the right part. 3. Combine the parts: 110 and 21. So, \( 103 \times 107 = 11021 \). This Vedic method provides a quick way to calculate such products.In simple words: For 103 and 107, the first part '10' is the same, and the last digits '3' and '7' add up to 10. So, we multiply 10 by (10+1) which gives 110. Then we multiply 3 by 7, which gives 21. Put these two parts together to get 11021.

🎯 Exam Tip: When using this sutra, make sure the right-hand part of the answer (product of last digits) always has as many digits as the number of zeros in the base (e.g., two digits for base 100, three for base 1000). If it's less, add leading zeros.

 

Question 11. 294 x 206
Answer: We apply the Ekadhikena Purvena sutra to multiply \( 294 \times 206 \). Here, the digits before the unit place are not strictly 'same' (29 and 20). This application of Ekadhikena Purvena is slightly different, often used for numbers near a base like 200 or 300 where the first digits are close or the same base is used. This method can also be used if the *first parts* (e.g., '2' from 294 and 206) are common, and the *remaining parts* (94 and 06) sum to 100. 1. The common initial digit is 2. Multiply this by 'one more than itself': \( 2 \times (2+1) = 2 \times 3 = 6 \). This is the left part. 2. Multiply the remaining parts: \( 94 \times 06 = 564 \). This forms the right part. 3. Combine the parts: 6 and 564. Since we are dealing with numbers close to 200, the right part needs to have a specific number of digits (typically two if the base is 100, here 3 for 564). So, \( 294 \times 206 = 60564 \). This sutra allows quick calculation for numbers with a common first part and complements in the remaining parts.In simple words: For 294 and 206, the first digit is 2. Multiply 2 by (2+1), which is 6. Then multiply the remaining parts, 94 and 06, which gives 564. Put them together to get 60564.

🎯 Exam Tip: For numbers where the last digits sum to 100 (or other power of 10), ensure the product of the last digits always occupies two (or more) places. Add leading zeros if necessary (e.g., \( 6 \times 4 = 24 \), write as \( 024 \)).

 

Question 12. 413 x 487
Answer: We apply the Ekadhikena Purvena sutra to multiply \( 413 \times 487 \). The common first digits are 4. The last two digits sum to 100: \( 13 + 87 = 100 \). 1. Multiply the common first digit by 'one more than itself': \( 4 \times (4+1) = 4 \times 5 = 20 \). This is the left part. 2. Multiply the remaining two-digit numbers: \( 13 \times 87 = 1131 \). This is the right part. 3. Combine the parts: 20 and 1131. So, \( 413 \times 487 = 201131 \). This method is very useful when two parts of the numbers satisfy specific conditions.In simple words: For 413 and 487, the first digit '4' is the same. The last two digits '13' and '87' add up to 100. Multiply 4 by (4+1), which is 20. Then multiply 13 by 87, which is 1131. Combine these to get 201131.

🎯 Exam Tip: This sutra is adaptable for multiplying numbers where the first parts are identical and the remaining parts are complements (sum to 10, 100, 1000, etc.).

Multiply with the help of Sutra Ekanyunena Purvena

 

Question 13. 54 x 99
Answer: The Ekanyunena Purvena sutra is a Vedic multiplication method particularly useful when one of the numbers is a series of nines (like 9, 99, 999, etc.). For \( 54 \times 99 \): 1. The left-hand side (L.H.S) of the answer is the first number decreased by one: \( 54 - 1 = 53 \). 2. The right-hand side (R.H.S) is the series of nines minus the L.H.S: \( 99 - 53 = 46 \). 3. Combine the L.H.S and R.H.S to get the final answer. So, \( 54 \times 99 = 5346 \). This method efficiently calculates products by breaking down the multiplication into simpler subtraction and concatenation.In simple words: To multiply 54 by 99, first take 1 away from 54, which gives 53. This is the first part of the answer. Then, subtract 53 from 99, which gives 46. This is the second part. Put them together to get 5346.

🎯 Exam Tip: Ensure the number of digits in the multiplier (the number with nines) is equal to or greater than the number of digits in the multiplicand for this sutra to apply directly. If the multiplier has more digits (e.g., \( 54 \times 999 \)), you might need to add leading zeros to the first number (e.g., \( 054 \times 999 \)).

 

Question 14. 214 x 999
Answer: We use the Ekanyunena Purvena sutra for \( 214 \times 999 \). 1. The left-hand side (L.H.S) of the answer is the first number decreased by one: \( 214 - 1 = 213 \). 2. The right-hand side (R.H.S) is the series of nines minus the L.H.S: \( 999 - 213 = 786 \). 3. Combine the L.H.S and R.H.S to get the final answer. So, \( 214 \times 999 = 213786 \). This method is very fast for multiplying any number by an equal number of nines.In simple words: To multiply 214 by 999, first subtract 1 from 214 to get 213. This is the left part. Then subtract 213 from 999 to get 786. This is the right part. Put them together for the answer: 213786.

🎯 Exam Tip: This sutra is particularly efficient when the number of digits in the multiplicand matches the number of nines in the multiplier.

 

Question 16. 342 x 99999
Answer: We use the Ekanyunena Purvena sutra for \( 342 \times 99999 \). Since the number of nines (5 digits) is more than the multiplicand's digits (3 digits), we add leading zeros to the multiplicand to match the length. So, we consider \( 00342 \times 99999 \). 1. The left-hand side (L.H.S) is the first number decreased by one: \( 00342 - 1 = 00341 \). 2. The right-hand side (R.H.S) is the series of nines minus the L.H.S: \( 99999 - 00341 = 99658 \). 3. Combine the L.H.S and R.H.S. So, \( 342 \times 99999 = 0034199658 = 34199658 \). This demonstrates how to use the sutra when the multiplier has more nines than the multiplicand's digits.In simple words: For 342 multiplied by 99999, first think of 342 as 00342. Subtract 1 from it to get 00341. This is the first part. Then subtract 00341 from 99999 to get 99658. Put them together: 34199658.

🎯 Exam Tip: When the number of nines is greater than the digits in the other number, prefix zeros to the smaller number to make their digit counts equal before applying the sutra. This ensures correct place value alignment in the right-hand part of the answer.

 

Question 17. 73 x 9
Answer: We use the Ekanyunena Purvena sutra for \( 73 \times 9 \). 1. The left-hand side (L.H.S) of the answer is the first number decreased by one: \( 73 - 1 = 72 \). 2. The right-hand side (R.H.S) of the answer is the series of nines minus the L.H.S. However, here, the '9' is a single digit, and 72 is two digits. This specific variation of Ekanyunena Purvena for a smaller number of nines works differently: We have \( 73 \times 9 \). We can write this as \( 73 \times (10 - 1) = 730 - 73 \). However, the Vedic method given follows a pattern: L.H.S \( = 73 - 1 = 72 \). R.H.S \( = 9 - 72 \) (This step indicates a specific Vedic process, possibly involving complements or a base shift). The provided calculation proceeds to: \( 729 - 72 = 657 \). This implies an alternative interpretation where the L.H.S `72` and `9` are combined (`729`), and then `72` (the L.H.S) is subtracted. So, \( 73 \times 9 = 657 \). This demonstrates a flexible application of Vedic principles beyond simple concatenation.In simple words: For 73 multiplied by 9, first take 1 away from 73 to get 72. Then, combine 72 and 9 to form 729. From this, subtract 72 to get 657.

🎯 Exam Tip: For multiplying by a single 9, a common approach is to multiply by 10 and then subtract the original number (e.g., \( 73 \times 9 = 73 \times 10 - 73 = 730 - 73 = 657 \)). This provides a good cross-check for the Vedic method.

 

Question 18. 467 x 99
Answer: We use the Ekanyunena Purvena sutra for \( 467 \times 99 \). Since the number of digits in the multiplicand (467) is more than the number of nines (99), we apply a slightly modified approach. 1. The left-hand part of the answer is \( 467 - 1 = 466 \). 2. The calculation then proceeds: \( 467 \times 99 = 467 - 1 / 99 - 466 \) (This usually means \( (467-1) \) followed by \( (99 - (complement of 467 \text{ to } 100)) \)). A common method for this case is to multiply the number by 100, then subtract the number: \( 467 \times 99 = 467 \times (100 - 1) = 46700 - 467 = 46233 \). The provided steps of \( 466 / 99 - 466 \) and then \( 46699 - 466 \) lead to the correct answer. \( 46699 - 466 = 46233 \). So, \( 467 \times 99 = 46233 \). This adaptation of the Ekanyunena Purvena principle is very efficient for such multiplications.In simple words: For 467 multiplied by 99, first take 1 away from 467 to get 466. Then, place 99 after 466 to make 46699. From this number, subtract 466 to get the final answer: 46233.

🎯 Exam Tip: When the multiplicand has more digits than the string of nines, another simple Vedic method is to subtract 1 from the multiplicand to get the leftmost part, then write the nines string, and subtract the leftmost part from that combined number (e.g., \( N \times 99 = (N-1)00 + (100 - N_{\text{last 2 digits}}) \)).

By Vedic Method Multiply the Following:

 

Question 19. Multiply \( 15\frac{5}{7} \times 15\frac{2}{7} \) using Ekadhikena Purvena.
Answer: We apply the Ekadhikena Purvena sutra for multiplying mixed numbers when the integer parts are the same and the fractional parts sum to 1. Here, the integer parts are both 15. The fractional parts are \( \frac{5}{7} \) and \( \frac{2}{7} \), and their sum is \( \frac{5}{7} + \frac{2}{7} = \frac{7}{7} = 1 \). 1. Multiply the integer part by 'one more than itself': \( 15 \times (15+1) = 15 \times 16 = 240 \). This is the left part. 2. Multiply the fractional parts: \( \frac{5}{7} \times \frac{2}{7} = \frac{10}{49} \). This is the right part. 3. Combine the parts: 240 and \( \frac{10}{49} \). So, \( 15\frac{5}{7} \times 15\frac{2}{7} = 240\frac{10}{49} \). This method is very elegant for certain types of fractional multiplications.In simple words: For these numbers, the whole number part (15) is the same, and the fractions (\( \frac{5}{7} \) and \( \frac{2}{7} \)) add up to 1. So, multiply 15 by (15+1), which is 240. Then multiply the two fractions, which is \( \frac{10}{49} \). Put them together as 240 and \( \frac{10}{49} \).

🎯 Exam Tip: Always verify that the fractional parts sum exactly to 1 before applying this specific variation of the Ekadhikena Purvena sutra for mixed number multiplication.

 

Question 20. \( 24\frac{10}{13} \times 24\frac{3}{13} \)
Answer: We use the Ekadhikena Purvena sutra for multiplying these mixed fractions. The integer parts are both 24. The fractional parts are \( \frac{10}{13} \) and \( \frac{3}{13} \), and their sum is \( \frac{10}{13} + \frac{3}{13} = \frac{13}{13} = 1 \). 1. Multiply the integer part by 'one more than itself': \( 24 \times (24+1) = 24 \times 25 = 600 \). This is the left part. 2. Multiply the fractional parts: \( \frac{10}{13} \times \frac{3}{13} = \frac{30}{169} \). This is the right part. 3. Combine the parts: 600 and \( \frac{30}{169} \). So, \( 24\frac{10}{13} \times 24\frac{3}{13} = 600\frac{30}{169} \). This method makes calculations quick and easy for suitable mixed numbers.In simple words: The whole number (24) is the same, and the fractions (\( \frac{10}{13} \) and \( \frac{3}{13} \)) add up to 1. So, multiply 24 by (24+1), which is 600. Then multiply the fractions (\( \frac{10}{13} \times \frac{3}{13} \)) to get \( \frac{30}{169} \). The answer is 600 and \( \frac{30}{169} \).

🎯 Exam Tip: This Vedic method saves time compared to converting mixed numbers to improper fractions and then multiplying, provided the conditions for the sutra are met.

 

Question 21. 4.5 x 4.5
Answer: We use the Ekadhikena Purvena sutra for multiplying numbers with decimal parts. The integer parts are both 4. The decimal parts are 0.5 and 0.5, and their sum is \( 0.5 + 0.5 = 1 \). 1. Multiply the integer part by 'one more than itself': \( 4 \times (4+1) = 4 \times 5 = 20 \). This is the left part. 2. Multiply the decimal parts: \( 0.5 \times 0.5 = 0.25 \). This is the right part. 3. Combine the parts: 20 and 0.25. So, \( 4.5 \times 4.5 = 20.25 \). This method is applicable to decimals where the whole number parts are identical and the fractional parts (decimals) sum to 1.In simple words: For 4.5 multiplied by 4.5, the whole number (4) is the same. The decimal parts (0.5 and 0.5) add up to 1. So, multiply 4 by (4+1), which is 20. Then multiply 0.5 by 0.5, which is 0.25. Combine these to get 20.25.

🎯 Exam Tip: Remember to correctly place the decimal point in the final answer based on the total number of decimal places in the original numbers (here, one decimal place in each number, so two in the product).

 

Question 22. 9.85 x 9.15
Answer: We use the Ekadhikena Purvena sutra for multiplying numbers with decimal parts. The integer parts are both 9. The decimal parts are 0.85 and 0.15, and their sum is \( 0.85 + 0.15 = 1.00 \). 1. Multiply the integer part by 'one more than itself': \( 9 \times (9+1) = 9 \times 10 = 90 \). This is the left part. 2. Multiply the decimal parts: \( 0.85 \times 0.15 = 0.1275 \). This is the right part. 3. Combine the parts: 90 and 0.1275. So, \( 9.85 \times 9.15 = 90.1275 \). This Vedic method simplifies the multiplication of numbers with complementary decimal parts.In simple words: For 9.85 multiplied by 9.15, the whole number (9) is the same. The decimal parts (0.85 and 0.15) add up to 1. So, multiply 9 by (9+1), which is 90. Then multiply 0.85 by 0.15, which is 0.1275. Combine these to get 90.1275.

🎯 Exam Tip: When multiplying decimal parts, ensure you count the total number of decimal places in the original numbers to correctly place the decimal point in their product.