onboardingMrs Lee has some pencils. If she packs them into bundles of 5, she will have 109 extra pencils. If she packs them into bundles of 9, she will need 7 more pencils. How many pencils are there?
Question
onboardingMrs Lee has some pencils. If she packs them into bundles of 5, she will have 109 extra pencils. If she packs them into bundles of 9, she will need 7 more pencils. How many pencils are there?
Solution
Let's solve this problem step by step:
Step 1: Let's denote the number of pencils Mrs. Lee has as X.
Step 2: According to the problem, if Mrs. Lee packs the pencils into bundles of 5, she will have 109 extra pencils. This can be written as the equation X = 5n + 109, where n is the number of bundles.
Step 3: The problem also states that if she packs the pencils into bundles of 9, she will need 7 more pencils. This can be written as the equation X = 9m - 7, where m is the number of bundles.
Step 4: Now we have a system of two equations, and we can solve it by equating the two expressions for X:
5n + 109 = 9m - 7
Step 5: Rearrange the equation to isolate m:
9m = 5n + 116
m = (5n + 116) / 9
Step 6: Since m and n must be integers (as they represent the number of bundles), (5n + 116) must be a multiple of 9. By trying different values of n, we find that n = 16 satisfies this condition.
Step 7: Substitute n = 16 into the first equation to find X:
X = 5 * 16 + 109 = 189
So, Mrs. Lee has 189 pencils.
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