The table shows Jim’s exercise times over the fi rst 5 days: Day
0 1 2 3 4 5 Exercise time (min) 5 7 9 11 13 15
A word formula, we could use to describe Jim’s exercise time is: Time spent each day = 5 minutes at the start of his regime + 2 minutes multiplied by the number of days This can be written more simply as a mathematical formula:
T = 5 + 2(D)
This represents the fi nal value of time spent (this is a variable)
This is the starting amount of time spent (this is a constant)
This represents the extra minutes added per day (this is the rate of change)
This represents the number of days, or the stage of the pattern (this is a variable)
We can use this formula to quickly fi nd how much time Jim spends exercising on any particular day. For example, how many minutes did Jim spend exercising on day 12? In this case, the number of days is 12, so we put ‘12’ in as the D value. T = 5 + 2(D) T = 5 + 2(12) T = 5 + 24 T = 29 minutes
In general, a formula for a linear pattern can be formed as: Value (total) = starting value + rate of change (stage number)
Worked example Holly is saving for a new phone. She received a gift of €50 and plans to save €20 per week. (i) Create a table showing how much money Holly will have in her savings after fi ve weeks. (ii) Describe the pattern in words: Total amount in savings = _____________________ (iii) In the table, identify the variables and the constant.
(iv) Using the sentence you created in part (ii), create a formula for the pattern. Explain what each letter you have used stands for.
(v) Use your formula to fi nd out how much money Holly will have after six weeks.
(vi) If the phone costs €210, use trial and improvement to fi nd out how long it will take for Holly to save for the phone.