Problems and Solutions on Integers
Problems related to integer numbers in mathematics are presented along with their solutions.
Problem 1: Find two consecutive integers whose sum is equal 135.
Solution to Problem 1:
Let $x$ and $x + 1$ (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 135 to write the equation: \begin{align} x + (x + 1) &= 135 \\ 2x+1 &= 135 \\ 2x &= 134 \\ x &= 134/2 \\ x &= 67 \end{align} So, the two numbers are $x = 67$ and $x + 1 = 68$
Problem 2: Find three consecutive integers whose sum is equal to 366.
Solution to Problem 2:
Let the three numbers be $x$, $x + 1$ and $x + 2$. their sum is equal to 366, hence: $$x + (x + 1) + (x + 2) = 366$$ Solve for $x$ and find the three numbers: $x = 121$ , $x + 1 = 122$ and $x + 2 = 123$
Problem 3: The sum of three consecutive even integers is equal to 84. Find the numbers.
Solution to Problem 3:
The difference between two even integers is equal to 2. let $x$, $x + 2$ and $x + 4$ be the three numbers. Their sum is equal to 84, hence $$x + (x + 2) + (x + 4) = 84$$ Solve for $x$ and find the three numbers: $x = 26$, $x + 2 = 28$ and $x + 4 = 30$.
Problem 4: The sum on an odd integer and twice its consecutive is equal to equal to 3757. Find the number
Solution to Problem 4:
The difference between two odd integers is equal to 2. let $x$ be an odd integer and $x + 2$ be its consecutive. The sum of $x$ and twice its consecutive is equal to 3757 gives an equation of the form. $$x + 2(x + 2) = 3757 \\ x=1251$$
Problem 5: The sum of the first and third of three consecutive odd integers is 131 less than three times the second integer. Find the three integers.
Solution to Problem 5:
Let $x$, $x + 2$ and $x + 4$ be three integers. The sum of the first $x$ and third $x + 4$ is given by $x + (x + 4)$
131 less than three times the second $3(x + 2)$ is given by $3(x + 2) - 131$
"The sum of the first and third is 131 less than three times the second" gives $$x + (x + 4) = 3(x + 2) - 131$$ Solve for $x$ and find all three numbers: $x = 129$ , $x + 2 = 131$ , $x + 4 = 133$
Problem 6: The product of two consecutive odd integers is equal to 675. Find the two integers.
Solution to Problem 6:
Let $x$, $x + 2$ be the two integers. Their product is equal to 675 $$x (x + 2) = 675$$
Expand to obtain a quadratic equation. \begin{align} x^2 + 2 x - 675 =0 \end{align} Solve for x to obtain two solutions
$x = 25$ or $x = -27$
if $x = 25$ then $x + 2 = 27$
if $x = -27$ then $x + 2 = -25$
We have two solutions. The two numbers are either 25 and 27 or -27 and -25.
Problem 7: Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two is equal to 742.
Solution to Problem 7:
Let $x$, $x + 2$, $x + 4$ and $x + 6$ be the four integers. The sum of the first two $x + (x + 2)$
twice the sum of the last two is written as $$2 ((x + 4) + (x + 6)) = 4 x + 20$$ sum of the first two added to twice the sum of the last two is equal to 742 is written as $$x + (x + 2) + 4 x + 20 = 742$$ Solve for $x$ and find all four numbers: $x=120$ , $x + 2 = 122$ , $x + 4 = 124$ , $x + 6 = 126$
Problem 8: When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 729 more than four times the middle integer. Find the numbers and check your answer.
Solution to Problem 8:
Let $x$, $x + 2$ and $x + 4$ be the three integers. "The smallest added to four times the largest is written as follows"
$$x + 4 (x + 4)$$ "729 more than four times the middle integer" is written as follows $$729 + 4 (x + 2)$$ "When the smallest is added to four times the largest, it produces a result 729 more than four times the middle" is written as follows $$x + 4 (x + 4) = 729 + 4 (x + 2)$$ Solve for x and find all three numbers \begin{align} x + 4 x + 16 &= 729 + 4 x + 8 \\ x &= 721 \\ x + 2 &= 723 \\ x + 4 &= 725 \end{align}
Problem 1: Find two consecutive integers whose sum is equal 135.
Solution to Problem 1:
Let $x$ and $x + 1$ (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 135 to write the equation: \begin{align} x + (x + 1) &= 135 \\ 2x+1 &= 135 \\ 2x &= 134 \\ x &= 134/2 \\ x &= 67 \end{align} So, the two numbers are $x = 67$ and $x + 1 = 68$
Problem 2: Find three consecutive integers whose sum is equal to 366.
Solution to Problem 2:
Let the three numbers be $x$, $x + 1$ and $x + 2$. their sum is equal to 366, hence: $$x + (x + 1) + (x + 2) = 366$$ Solve for $x$ and find the three numbers: $x = 121$ , $x + 1 = 122$ and $x + 2 = 123$
Problem 3: The sum of three consecutive even integers is equal to 84. Find the numbers.
Solution to Problem 3:
The difference between two even integers is equal to 2. let $x$, $x + 2$ and $x + 4$ be the three numbers. Their sum is equal to 84, hence $$x + (x + 2) + (x + 4) = 84$$ Solve for $x$ and find the three numbers: $x = 26$, $x + 2 = 28$ and $x + 4 = 30$.
Problem 4: The sum on an odd integer and twice its consecutive is equal to equal to 3757. Find the number
Solution to Problem 4:
The difference between two odd integers is equal to 2. let $x$ be an odd integer and $x + 2$ be its consecutive. The sum of $x$ and twice its consecutive is equal to 3757 gives an equation of the form. $$x + 2(x + 2) = 3757 \\ x=1251$$
Problem 5: The sum of the first and third of three consecutive odd integers is 131 less than three times the second integer. Find the three integers.
Solution to Problem 5:
Let $x$, $x + 2$ and $x + 4$ be three integers. The sum of the first $x$ and third $x + 4$ is given by $x + (x + 4)$
131 less than three times the second $3(x + 2)$ is given by $3(x + 2) - 131$
"The sum of the first and third is 131 less than three times the second" gives $$x + (x + 4) = 3(x + 2) - 131$$ Solve for $x$ and find all three numbers: $x = 129$ , $x + 2 = 131$ , $x + 4 = 133$
Problem 6: The product of two consecutive odd integers is equal to 675. Find the two integers.
Solution to Problem 6:
Let $x$, $x + 2$ be the two integers. Their product is equal to 675 $$x (x + 2) = 675$$
Expand to obtain a quadratic equation. \begin{align} x^2 + 2 x - 675 =0 \end{align} Solve for x to obtain two solutions
$x = 25$ or $x = -27$
if $x = 25$ then $x + 2 = 27$
if $x = -27$ then $x + 2 = -25$
We have two solutions. The two numbers are either 25 and 27 or -27 and -25.
Problem 7: Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two is equal to 742.
Solution to Problem 7:
Let $x$, $x + 2$, $x + 4$ and $x + 6$ be the four integers. The sum of the first two $x + (x + 2)$
twice the sum of the last two is written as $$2 ((x + 4) + (x + 6)) = 4 x + 20$$ sum of the first two added to twice the sum of the last two is equal to 742 is written as $$x + (x + 2) + 4 x + 20 = 742$$ Solve for $x$ and find all four numbers: $x=120$ , $x + 2 = 122$ , $x + 4 = 124$ , $x + 6 = 126$
Problem 8: When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 729 more than four times the middle integer. Find the numbers and check your answer.
Solution to Problem 8:
Let $x$, $x + 2$ and $x + 4$ be the three integers. "The smallest added to four times the largest is written as follows"
$$x + 4 (x + 4)$$ "729 more than four times the middle integer" is written as follows $$729 + 4 (x + 2)$$ "When the smallest is added to four times the largest, it produces a result 729 more than four times the middle" is written as follows $$x + 4 (x + 4) = 729 + 4 (x + 2)$$ Solve for x and find all three numbers \begin{align} x + 4 x + 16 &= 729 + 4 x + 8 \\ x &= 721 \\ x + 2 &= 723 \\ x + 4 &= 725 \end{align}
Post a Comment for "Problems and Solutions on Integers"