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Welcome to Mind Over Math

MIND OVER MATH is an organization designed to help students to succeed in mathematics through homework help, personal tutoring, exam reviews and prep programs. We help students in late elementary school and throughout high school. We can also use our Personal Tutoring to help those in the early grades and those out of high school.

We sell tools for success. We help students get over the hurdle of mathematics and help them to see the beauty of the subject.”
– Kevin Downe, President of Mind Over Math Inc.

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How sharp was your thinking? Here's MOM's answer to this week's problem! ...

We want to find a number of pears (p) and a number of apples (a) which, when their prices are combined together, have a total of $1.43. This is equivalent to solving the equation 17p+15a=143 over the integers. One way to solve this is to check and see how many times you have to take 17 off of 143 before you get to a multiple of 15, as multiples of 15 are easy to recognize below 150. Using this method, you can easily find that you need 5 apples and 4 pears.

...There is a more theoretical approach to this problem:

The greatest common divisor of 17 and 15 is 1, as 17 is prime. Using the division algorithm for finding the greatest common divisor of two numbers, you can get 17=(1)15+2 and 15=(7)2+1, and 2=(2)1+0. From this we can conclude that 1 is the greatest common divisor, and that 2=17-15 and 1=15-7(2), so 1=15-7(17-15), or 1=(8)15-(7)17. This related to our solution. If we multiply by 143 at this stage, we can see that 143=-(1001)17+(1144)15. We now need to get either variable into an acceptable range using the fact that for a given solution (in this case (-1001,1144)), you can find any other one using the formula p=-1001+15t, a=1144-17t. To find the smallest value of t that gives reasonable positive answers, we can see that 1144/17=67 remainder 5. For this reason, we chose t=67, and get the result, p=4, a=5, as we did in the first part.
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