If the sum of two integers is -21 and one of them is -10 then the other is
-32
32
-11
11
Answer
Other integer = sum of two integers − (the given integer)
= −21 − (−10)
= −21 + 10
= −(21 − 10)
= −11.
Hence, option 3 is the correct option.
The number of natural numbers between the smallest natural number and the greatest 2-digit number is
90
97
98
99
Answer
The smallest natural number is 1.
The greatest 2-digit number is 99.
The natural numbers strictly between 1 and 99 are:
2, 3, 4, ..., 98.
Counting them gives 98 − 2 + 1 = 97 natural numbers.
Hence, option 2 is the correct option.
Find the value of 25 × 37 × 8 × 6 by suitable arrangement.
Answer
We rearrange the factors using the commutative and associative properties of multiplication so as to group the numbers whose product is easy to compute.
Note that 25 × 8 = 200, which is easy to handle.
25 × 37 × 8 × 6 = (25 × 8) × (37 × 6)
= 200 × 222
= 44400.
Hence, 25 × 37 × 8 × 6 = 44400.
Write four consecutive integers preceding -97.
Answer
Four consecutive integers preceding (i.e. just before) −97 are obtained by repeatedly subtracting 1 from −97.
−97 − 1 = −98
−98 − 1 = −99
−99 − 1 = −100
−100 − 1 = −101
Listing them in order from smallest to largest, the four consecutive integers preceding −97 are −101, −100, −99 and −98.
Hence, the four consecutive integers preceding −97 are −101, −100, −99 and −98.
Write the greatest and the smallest 4-digit numbers using four different digits with the condition that 5 occurs at ten's place.
Answer
A 4-digit number has four places: thousands, hundreds, tens and ones. Since 5 is fixed at the ten's place, the number has the form:
| Thousands | Hundreds | Tens | Ones |
|---|---|---|---|
| ? | ? | 5 | ? |
The remaining three digits must be different from each other and different from 5.
Greatest 4-digit number: To make the number greatest, we put the largest available digits at the higher places (left side).
- Thousands place: 9 (the largest digit)
- Hundreds place: 8 (next largest, ≠ 9 and ≠ 5)
- Tens place: 5 (fixed)
- Ones place: 7 (next largest, ≠ 9, ≠ 8 and ≠ 5)
So the greatest such number is 9857.
Smallest 4-digit number: To make the number smallest, we put the smallest available digits at the higher places. However, the thousands digit cannot be 0 (otherwise it will not remain a 4-digit number).
- Thousands place: 1 (smallest non-zero digit)
- Hundreds place: 0 (smallest digit, ≠ 1 and ≠ 5)
- Tens place: 5 (fixed)
- Ones place: 2 (smallest digit ≠ 0, ≠ 1 and ≠ 5)
So the smallest such number is 1052.
Hence, the greatest 4-digit number is 9857 and the smallest 4-digit number is 1052.
Write all possible natural numbers formed by the digits 7, 0 and 3. Repetition of digits is not allowed.
Answer
We have to form natural numbers using each of the digits 7, 0 and 3 exactly once.
The one-digit numbers that can be formed using 7, 0, 3 are 7, 0, 3.
2-digit numbers that can be formed using 7, 0, 3 are:
30, 37, 70, 73 (0 cannot be on the tens place because then the number will become 1-digit)
Now we are forming 3-digit numbers.
The hundreds digit cannot be 0, because then the number would not be a 3-digit number. So the hundreds digit must be either 7 or 3.
Case I: Hundreds digit is 7.
The remaining digits 0 and 3 can be arranged at the tens and ones places in 2 ways:
- 7 0 3 → 703
- 7 3 0 → 730
Case II: Hundreds digit is 3.
The remaining digits 0 and 7 can be arranged at the tens and ones places in 2 ways:
- 3 0 7 → 307
- 3 7 0 → 370
Hence, all the possible natural numbers formed by the digits 7, 0 and 3 (without repetition) are 3, 7, 30, 37, 70, 73, 307, 370, 703 and 730.
Find the value of: -237 - (-328) + (-205) - 76 + 89
Answer
We first change the sign of each integer being subtracted and then add. After that, we group the positive and negative integers separately.
−237 − (−328) + (−205) − 76 + 89
= −237 + 328 − 205 − 76 + 89
Now group the positive and negative integers:
Sum of positive integers = 328 + 89 = 417.
Sum of absolute values of negative integers = 237 + 205 + 76 = 518.
Therefore,
−237 + 328 − 205 − 76 + 89 = 417 − 518
= −(518 − 417)
= −101.
Hence, −237 − (−328) + (−205) − 76 + 89 = −101.
Abhijeet's school is 3 km 520 m away from his home. One day while returning from his school, just after covering 1 km 370 m distance, he saw a woman who was bleeding. He took her to the nearest hospital which was 2 km 775 m away from that place and got her admitted. He came back to his home which was 4 km 565 m from the hospital. Find the distance covered by Abhijeet on that day.
Answer
On that day, Abhijeet covered the following distances one after another:
(1) From home to school (in the morning) = 3 km 520 m.
(2) From school, on his way back, until he saw the woman = 1 km 370 m.
(3) From that place to the nearest hospital = 2 km 775 m.
(4) From the hospital back to his home = 4 km 565 m.
Total distance covered = 3 km 520 m + 1 km 370 m + 2 km 775 m + 4 km 565 m.
Converting each distance into metres:
3 km 520 m = 3520 m
1 km 370 m = 1370 m
2 km 775 m = 2775 m
4 km 565 m = 4565 m
Adding:
3520 + 1370 = 4890 m
2775 + 4565 = 7340 m
4890 + 7340 = 12230 m
Now 12230 m = 12000 m + 230 m = 12 km 230 m.
Hence, the total distance covered by Abhijeet on that day is 12 km 230 m (i.e. 12230 m).
Arrange the following integers in descending order:
-353, 207, -289, 702, -335, 0, -77
Answer
Descending order means arranging from the greatest to the smallest.
The given integers are −353, 207, −289, 702, −335, 0, −77.
The positive integers are 207 and 702. In decreasing order: 702 > 207.
Then comes 0, which is greater than every negative integer.
The negative integers are −353, −289, −335 and −77. Their absolute values are 353, 289, 335 and 77.
Arranging the absolute values in decreasing order: 353 > 335 > 289 > 77. So, for these negative integers, the one with bigger absolute value is smaller. Hence the negatives in decreasing order are −77 > −289 > −335 > −353.
Combining: 702 > 207 > 0 > −77 > −289 > −335 > −353.
Hence, the given integers in descending order are 702, 207, 0, −77, −289, −335, −353.
Find the smallest five-digit number which is exactly divisible by 254.
Answer
The smallest five-digit number is 10000.
On dividing 10000 by 254, we get:
254 × 39 = 9906
254 × 40 = 10160
So 10000 = 254 × 39 + 94, i.e. the remainder is 94.
This means 10000 is not exactly divisible by 254. To get the smallest five-digit number which is exactly divisible by 254, we add (254 − 94) = 160 to 10000 (so that the next multiple of 254 is reached):
10000 + 160 = 10160.
Check: 10160 ÷ 254 = 40, and 254 × 40 = 10160. So 10160 is a 5-digit number and is exactly divisible by 254.
Hence, the smallest five-digit number which is exactly divisible by 254 is 10160.