This problem solving activity has an algebra focus.
Tia and Tom are twins. Tia saves and Tom spends.
Tom finds a $20 note on Sunday evening and spends $2 a day starting on Monday.
Coincidentally, Tia starts work on Monday and gets $2.50 a day.
How long will it be before Tia has more money than Tom?
This problem involves students finding a mathematical solution for a practical problem. This can be done in a number of ways, including guess and check, and using a table. However, both a graphical approach and an algebraic one are explored in more depth here. This is done to enable students to see the value of these approaches, and how they can be effectively applied to other situations. To work successfully with this problem, your students should have some knowledge of working with continuous number sequences, linear rules graphs, and tables.
Tia and Tom are twins. Tia saves and Tom spends. Tom finds a $20 note on Sunday evening and spends $2 a day starting on Monday. Coincidentally, Tia starts work on Monday and gets $2.50 a day. How long will it be before Tia has more money than Tom?
Write your own twins problem for others to solve.
In the graph below, we sketch the situation. Diamonds represent Tom’s finances and squares represent Tia’s situation.
From the graph we can easily see that the small squares go above the crosses on day 5, Friday. So it is five days until Tia has more money than Tom.
To do this algebraically we first have to find out the two rules for the number patterns produced by Tom and Tia’s activity. We can use the letter n to represent the number of days surpassed, and M to represent the total amount of money. Tom began with $20 and then spent $2 a day. So each day he had $2 less. His equation is
M = 20 – 2n
Tia earns $2.50 each day and so her equation is
M = 2.5n
Check these equations work by putting in actual values for n.
If the two Ms can ever be equal we will have:
2.5n = 20 – 2n, which becomes 4.5n = 20, so n is 4 and a bit.
If they are equal after 4 and a bit days, then one of them must have more for the first time on day 5. Since Tia’s money is increasing, she is just ahead on day 5.
Printed from https://nzmaths.co.nz/resource/topsy-turvy-twins at 4:39am on the 29th March 2024