Lesson Choose an unknown variable in a rational way for your problem
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<H2>Choose an unknown variable in a rational way for your problem</H2> <H3>Problem 1</H3>Tom had 4/5 as many stamps as Michael. After Michael gave away 3/7 of his stamps, Tom had 40 more stamps than Michael. How many stamps did Tom have? <B>Solution</B> <pre> Let 5x be the number of stamps Michael had originally. Then Tom had 4x stamps originally. After Michael gave away 3/7 of his stamps, you have THIS equation {{{(4/7)*(5x)}}} = 4x - 40. To solve, multiply both sides by 7. You will get 20x = 28x - 280 280 = 28x - 20x 280 = 8x x = 280/8 = 35. Tom had 4x = 4*35 = 140 stamps originally. <U>ANSWER</U> </pre> Notice that in my solution I introduced the unknown variable in a way to make my computations as simple as possible. <H3>Problem 2</H3>Mrs Johnson and her 2 sons Bernard and Raymond shared a sum of money. The ratio received by Mrs Johnson to that received by Bernard was 5 to 2 . The ratio of money received by Bernard to that received by Raymond was 3 to 4. After Mrs Johnson gave 20% of her share to Raymond and B ernard donated 1/3 of his share to charity, Raymond then had $4270 more than Bernard. What was the sum of money shared by the three of them. <B>Solution</B> <pre> From this statement, "The ratio received by Mrs Johnson to that received by Bernard was 5 to 2", I choose to write 5x for Mrs. Johnson amount and 2x for the Bernard amount. Here "x" is the common measure of these amounts, in dollars. From the second statement, "The ratio of money received by Bernard to that received by Raymond was 3 to 4" we conclude that Raymond then received {{{(4/3)*(2x)}}} = {{{(8x)/3}}} dollars. After re-distribution and donation, Raymond has {{{(8x)/3 + (1/5)*(5x)}}} = {{{(8x)/3 + x}}} = {{{(11x)/3}}} dollars; Bernard has {{{2x - (1/3)*(2x)}}} = {{{(2/3)*(2x)}}} = {{{(4x)/3}}}. Now Raymond has $4270 more than Bernard {{{(11x)/3}}} - {{{(4x)/3}}} = 4270 dollars. The setup is complete. Now we should solve this equation and find x. Multiply both sides by 3 11x - 4x = 3*4270 7x = 12810 x = 12810/7 = 1830. Thus the sum of money shared by the three of them was 5x + 2x + {{{(8x)/3}}} = {{{5*1830 + 2*1830 + (8*1830)/3}}} = 17690 dollars. <U>ANSWER</U>. The sum of money shared by the three of them was 17690 dollars. </pre> Solved completely, using only one single unknown variable, and in a way as it should be done and as it is expected to be done. My other additional lessons on solving single linear equations and word problems in one unknown in this site (section 2) are - <A HREF=https://www.algebra.com/algebra/homework/equations/One-more-lesson-on-solving-problems-by-the-backward-method.lesson>One more lesson on solving problems by the backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Solving-problems-in-three-unknowns-using-only-one-equation.lesson>Entertainment problems on finding three unknowns using only one equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Upper-level-word-problems-to-solve-using-single-linear-equation.lesson>Upper level word problems to solve using a single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/OVERVIEW-of-my-additional-lessons-on-solving-single-linear-equations-and-word-problems-in-one-unknown.lesson>OVERVIEW of my additional lessons on solving single linear equations and word problems in one unknown</A>
