Suppose that Diana expects to work for $40$ years and then retire, with a life expectancy of an additional $20$ years. Diana$$s earnings rise at a rate of $3$ percent per year in real terms, and the real interest rate is also $3$ percent. What constant fraction of income must she save in each working year to be able to finance a level of retirement income equal to $60$ percent of earnings in the final year prior to retirement? Show all the necessary steps.