problems on numbers chapter 4
Q1–10 of 50The sum of five consecutive odd numbers is 575. What is the sum of the next set of five consecutive odd numbers?
The average of five consecutive odd numbers is $\frac{575}{5} = 115$. Thus, the 5 numbers are $111, 113, 115, 117, 119$.
The next five consecutive odd numbers are $121, 123, 125, 127, 129$.
Their sum is $121 + 123 + 125 + 127 + 129 = 625$.
Correct Option: (d) None of these
The sum of three consecutive odd numbers and three consecutive even numbers together is 231. Also, the smallest odd number is 11 less than the smallest even number. What is the sum of the largest odd number and the largest even number?
Let the 3 consecutive odd numbers be $x, x+2, x+4$ (Sum = $3x+6$).
Let the 3 consecutive even numbers be $y, y+2, y+4$ (Sum = $3y+6$).
Given: $3x + 6 + 3y + 6 = 231 \implies 3(x+y) + 12 = 231 \implies x+y = 73$.
Given: $y - x = 11$.
Solving $x+y=73$ and $y-x=11$: $y = 42, x = 31$.
Largest odd number = $x+4 = 35$; Largest even number = $y+4 = 46$.
Sum of largest odd and even numbers = $35 + 46 = 81$.
Correct Option: (d) None of these (Value is 81)
Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is
Let the 3 consecutive odd numbers be $x, x+2, x+4$ (Sum = $3x+6$).
Let the 3 consecutive even numbers be $y, y+2, y+4$ (Sum = $3y+6$).
Given: $3x + 6 + 3y + 6 = 231 \implies 3(x+y) + 12 = 231 \implies x+y = 73$.
Given: $y - x = 11$.
Solving $x+y=73$ and $y-x=11$: $y = 42, x = 31$.
Largest odd number = $x+4 = 35$; Largest even number = $y+4 = 46$.
Sum of largest odd and even numbers = $35 + 46 = 81$.
Correct Option: (d) None of these (Value is 81)
The sum of four consecutive even integers is 1284. The greatest of them is
Let the numbers be $x, x+2, x+4, x+6$.
Sum: $4x + 12 = 1284 \implies 4x = 1272 \implies x = 318$.
The greatest integer is $x+6 = 318 + 6 = 324$.
Correct Option: (c) 324
The sum of three consecutive odd numbers is 20 more than the first of these numbers. What is the middle number?
Let three consecutive odd numbers be $x, x+2, x+4$.
Given: $(x + x+2 + x+4) = x + 20 \implies 3x + 6 = x + 20 \implies 2x = 14 \implies x = 7$.
Middle number is $x+2 = 7+2 = 9$.
Correct Option: (b) 9
The product of three consecutive even numbers when divided by 8 is 720. The product of their square roots is
Let numbers be $n-2, n, n+2$. Product is $(n-2)n(n+2) = 8 \times 720 = 5760$.
Test numbers: $16 \times 18 \times 20 = 5760$. So, the numbers are $16, 18, 20$.
Product of their square roots: $\sqrt{16} \times \sqrt{18} \times \sqrt{20} = 4 \times 3\sqrt{2} \times 2\sqrt{5} = 24\sqrt{10}$.
Correct Option: (b) $24\sqrt{10}$
The sum of three consecutive multiples of 3 is 72. What is the largest number?
Let the numbers be $3x, 3x+3, 3x+6$.
Sum: $9x + 9 = 72 \implies 9x = 63 \implies x = 7$.
Largest number is $3(7)+6 = 27$.
Correct Option: (c) 27
What is the sum of two consecutive even numbers, the difference of whose squares is 84?
Let numbers be $x$ and $x+2$.
Difference of squares: $(x+2)^2 - x^2 = 84 \implies 4x + 4 = 84 \implies 4x = 80 \implies x = 20$.
Numbers are $20$ and $22$. Sum = $20 + 22 = 42$.
Correct Option: (c) 42
The sum of the squares of three consecutive natural numbers is 2030. What is the middle number?
Let middle number be $x$. The numbers are $x-1, x, x+1$.
Sum of squares: $(x-1)^2 + x^2 + (x+1)^2 = 2030 \implies 3x^2 + 2 = 2030 \implies 3x^2 = 2028 \implies x^2 = 676 \implies x = 26$.
Correct Option: (b) 26
If the product of three consecutive integers is 120, then the sum of the integers is
Product of three consecutive integers = $120$.
Factoring $120$: $4 \times 5 \times 6 = 120$.
Sum of integers = $4 + 5 + 6 = 15$.
Correct Option: (d) 15