Find two consecutive odd integers such that the sum of four times the larger and twice the smaller is 194.
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n , n+2
4(n+2) + 2 n = 194
4 n + 8 + 2 n = 194
6 n = 186
n = 31
n+2 = 33
Sure! To solve this problem, let's break it down into steps:
Step 1: Let's assume the smaller odd integer is 'x'.
Step 2: Since we need to find two consecutive odd integers, we can say that the larger odd integer is 'x + 2'. This is because consecutive odd integers have a difference of 2.
Step 3: Now we have our two odd integers: 'x' and 'x + 2'.
Step 4: According to the problem statement, the sum of four times the larger odd integer and twice the smaller odd integer is 194. So we can write the equation as follows:
4(x + 2) + 2x = 194.
Step 5: Simplify the equation:
4x + 8 + 2x = 194.
6x + 8 = 194.
6x = 186.
Step 6: Solve for 'x':
x = 186/6.
x = 31.
Step 7: Now we can find the larger odd integer:
x + 2 = 31 + 2.
x + 2 = 33.
Step 8: So the two consecutive odd integers are 31 and 33.
To recap, we found that the two consecutive odd integers are 31 and 33. We solved the problem by setting up an equation using the sum of four times the larger odd integer and twice the smaller odd integer and then solving for 'x'.