The product of two consecutive positive integers is 90. Find the integers.
Let's use algebra and variables to solve this problem.
Let x be the first integer and x+1 be the second integer.
According to the problem, their product is 90, so:
x(x+1) = 90
Expanding the left side:
x^2 + x = 90
Now we have a quadratic equation that we can solve by factoring or using the quadratic formula. However, we can also guess and check since the factors of 90 are not too many.
Some possible pairs of factors that multiply to 90 are:
1 x 90 = 90
2 x 45 = 90
3 x 30 = 90
5 x 18 = 90
6 x 15 = 90
9 x 10 = 90
We want consecutive integers, so only the pair 9 x 10 works.
Therefore, x = 9 and x+1 = 10.
The two consecutive positive integers that have a product of 90 are 9 and 10.
I think in this case I would use common sense. 9 and 10 work instantly :)
Yes, that's correct! Sometimes using common sense or intuition can make the problem easier and faster to solve. In this case, recognizing that 9 and 10 are consecutive integers whose product is close to 90 could lead to the answer without having to use algebra or guess and check.
To solve this problem, we need to find two consecutive positive integers whose product is equal to 90.
Let's assume the first integer is n. The next consecutive integer would then be n+1.
The problem states that the product of these two integers is equal to 90, so we have the equation:
n * (n+1) = 90
To solve this equation, we can try different values of n until we find the solution that satisfies the equation.
Let's start by trying n = 9:
9 * (9+1) = 9 * 10 = 90
This satisfies the equation, so the two consecutive positive integers are 9 and 10.
Therefore, the solution is n = 9.
Thus, the two consecutive positive integers are 9 and 10.