there were 360 pumpkins in a pumpkin patch , but it was difficult for farmer alexander to find the perfect pumpkin.
evry third pumpkin was too small.
every fourth pumpkin was too green.
every fifth pumpkin had a broken stem.
every sixth pumpkin had the wrong shape.
How many perfect pumpkins did farmer alexander find in the pumpkin patch.
solve using equations
There are 144 perfect pumpkins.
To solve this problem, we can use the concept of the least common multiple (LCM) and solve it step by step.
Step 1: Find the LCM of 3, 4, 5, and 6.
The LCM of these four numbers is 60.
Step 2: Calculate how many pumpkins meet each condition.
To find the number of pumpkins that are too small, we divide 360 by 3 and round down since we are only interested in whole numbers. 360 ÷ 3 = 120.
To find the number of pumpkins that are too green, we divide 360 by 4 and round down. 360 ÷ 4 = 90.
To find the number of pumpkins with a broken stem, we divide 360 by 5 and round down. 360 ÷ 5 = 72.
To find the number of pumpkins with the wrong shape, we divide 360 by 6 and round down. 360 ÷ 6 = 60.
Step 3: Subtract the total number of pumpkins that meet any of the conditions from the total number of pumpkins in the patch to find the number of perfect pumpkins.
The number of pumpkins that meet any of the conditions is the sum of the numbers obtained in step 2: 120 + 90 + 72 + 60 = 342.
To find the number of perfect pumpkins, subtract this number from the total number of pumpkins in the patch: 360 - 342 = 18.
Therefore, farmer Alexander found 18 perfect pumpkins in the pumpkin patch.
To solve this problem using equations, we can use a method called the inclusion-exclusion principle. We will calculate the number of pumpkins that satisfy each condition individually, and then subtract the number of pumpkins that satisfy multiple conditions, as we don't want to count them multiple times.
Let's start by calculating the number of pumpkins that are too small. Every third pumpkin is too small, so we can set up the equation:
Number of pumpkins that are too small = 360 / 3 = 120
Next, we calculate the number of pumpkins that are too green. Every fourth pumpkin is too green, so the equation becomes:
Number of pumpkins that are too green = 360 / 4 = 90
For the broken stems, every fifth pumpkin has a broken stem:
Number of pumpkins with broken stems = 360 / 5 = 72
For the wrong shape, every sixth pumpkin has the wrong shape:
Number of pumpkins with wrong shape = 360 / 6 = 60
Now, we need to subtract the overlap cases – those that satisfy multiple conditions. To find the overlap between any two conditions, we need to find the least common multiple (LCM) of their denominators (3, 4, 5, and 6). In this case, the LCM is 60.
Let's consider the overlap between being too small and too green:
Number of pumpkins with both conditions = 360 / LCM(3, 4) = 360 / 12 = 30
Now let's calculate the overlap between having a broken stem and the wrong shape:
Number of pumpkins with both conditions = 360 / LCM(5, 6) = 360 / 30 = 12
Finally, we can add up these numbers:
Number of perfect pumpkins = Total number of pumpkins - (Number of pumpkins too small + Number of pumpkins too green + Number of pumpkins with broken stems + Number of pumpkins with wrong shape - Number of pumpkins with both conditions)
Number of perfect pumpkins = 360 - (120 + 90 + 72 + 60 - 30 - 12)
Number of perfect pumpkins = 144
Therefore, Farmer Alexander found 144 perfect pumpkins in the pumpkin patch.