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Math Lecture Notes
Investing

In-Class Activity link to here link to index

The Rule of 72

Doubling While Increasing

investment doubling chart

We have already discussed what interest is. We learned how to use the formulas for simple interest and compound interest.

Those formulas allowed us to find precise amounts of interest. Great!

However, in real life we sometimes want a quick and easy estimate about how much an investment might help us prepare for the future.

For the moment, let's use the compound interest formula to watch an investment grow. We can look for a pattern.

1. Imagine that you invest $1,000 in an account that earns 10% interest every year, compounded annually. How many years will it take to double your money?

1. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.11 = $1,100

After 2 Years = $1,000 × 1.12 = $1,210

After 3 Years = $1,000 × 1.13 = $1,331

After 4 Years = $1,000 × 1.14 = $1,464.10

After 5 Years = $1,000 × 1.15 = $1,610.51

After 6 Years = $1,000 × 1.16 ≈ $1,771.56

After 7 Years = $1,000 × 1.17 ≈ $1,948.72

After 8 Years = $1,000 × 1.18 ≈ $2,143.59

The original investment had almost doubled after 7 years, and has more than doubled after 8 years.

2. Imagine that you invest $1,000 in an account that earns 7% interest every year, compounded annually. How many years will it take to double your money?

2. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.071 = $1,070

After 2 Years = $1,000 × 1.072 = $1,144.90

After 3 Years = $1,000 × 1.073 ≈ $1,225.04

After 4 Years = $1,000 × 1.074 ≈ $1,310.80

After 5 Years = $1,000 × 1.075 ≈ $1,402.55

After 6 Years = $1,000 × 1.076 ≈ $1,500.73

After 7 Years = $1,000 × 1.077 ≈ $1,605.78

After 8 Years = $1,000 × 1.078 ≈ $1,718.19

After 9 Years = $1,000 × 1.079 ≈ $1,838.46

After 10 Years = $1,000 × 1.0710 ≈ $1,967.15

After 11 Years = $1,000 × 1.0711 ≈ $2,104.85

The original investment had almost doubled after 10 years, and has more than doubled after 11 years.

The Rule of 72 is a way to estimate how long it takes for doubling to happen.

The Rule of 72 (for Doubling)

The Rule of 72 says

annual interest rate × years until doubling ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

In our first example 10 rate × 7 years ≈ 72. The product 10 rate × 7 years was slightly below 72, hinting that the answer would happen soon after the seventh year.

In our second example 7 rate × 10 years ≈ 72. The product 7 rate × 10 years was slightly below 72, hinting that the answer would happen soon after the tenth year

3. Imagine that you invest $1,000 in an account that earns 2% interest every year, compounded annually. Use the Rule of 72 to estimate how many years will it take to double your money.

3. The Rule of 72 says 2 rate × years until doubling ≈ 72

We divide both sized by 2 to find an answer of 36 years.

A very long time! No one can save for retirement with only a 2% annual interest rate.

Halving While Decreasing

Does the Rule of 72 also work when an initial amount of money shrinks?

4. Imagine that you inherit $1,000. Each year you spend 10% of what is left. How many years will it take to halve your inheritance money?

4. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.1, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.1) = 0.9

After 1 Year = $1,000 × 0.91 = $900

After 2 Years = $1,000 × 0.92 = $810

After 3 Years = $1,000 × 0.93 = $729

After 4 Years = $1,000 × 0.94 = $656.10

After 5 Years = $1,000 × 0.95 = $590.49

After 6 Years = $1,000 × 0.96 ≈ $531.44

After 7 Years = $1,000 × 0.97 ≈ $478.30

The original investment had almost halved after 6 years, and has more than halved after 7 years.

Yes, the Rule of 72 again provided a close estimate. 10 rate × 7 years ≈ 72

Let's check again.

5. Imagine that you inherit $1,000. Each year you spend 7% of what is left. How many years will it take to halve your inheritance money?

5. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.07, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.07) = 0.93

After 1 Year = $1,000 × 0.931 = $930

After 2 Years = $1,000 × 0.932 = $864.90

After 3 Years = $1,000 × 0.933 ≈ $804.38

After 4 Years = $1,000 × 0.934 ≈ $748.05

After 5 Years = $1,000 × 0.935 ≈ $695.69

After 6 Years = $1,000 × 0.936 ≈ $646.99

After 7 Years = $1,000 × 0.937 ≈ $601.70

After 8 Years = $1,000 × 0.938 ≈ $559.58

After 9 Years = $1,000 × 0.939 ≈ $520.41

After 10 Years = $1,000 × 0.9310 ≈ $483.98

The original investment had almost halved after 9 years, and has more than halved after 10 years.

Yes, the Rule of 72 again provided a close estimate. 7 rate × 10 years ≈ 72

Hooray! The Rule of 72 works both ways.

The Rule of 72 (for Halving)

The Rule of 72 says

annual interest rate × years until halving ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

Facts link to here link to index

Mortgages

amortization table

Amortization is the paying off of debt with repeating, schedules repayments.

Imagine that you had a home or car loan, and needed to make the same monthly payment again and again for many years. What would be the proper monthly payment amount? This problem is tricky, because the debt shrinks with payments and grows with interest at the same time. But accountants have solved the problem many times, and saved the answers as an amortization table.

In this class we will use the amortization table to the right. In real life, the amortization table is huge. It has more rows, for interest rates with decimals (usually to the hundredths place). It has more columns, for loans with other durations of months and/or years.

Tangentially, there is a general rule for mortgages (or apartment payments). First, find 35% of your household's before-tax income. Second, find 25% of your household's after-tax income. Your household's total monthly expenses for housing should be between those two numbers.

If you are curious about local housing prices, check Zillow.

Knowing a Loan, Finding a Payment

The amortization table values show us the correct monthly payment for a $1,000 loan with a certain interest rate and duration.

We need to scale up this number by how many thousands are in the actual loan amount.

6. Find the monthly payment for a $130,000 mortgage that is a twenty-year loan with an interest rate of 6%.

6. The amortization table value for 6% and 20 years is $7.16.

So the monthly payment is $7.16 for each $1,000 of loan.

There are 130 thousands in the loan size of $130,000.

So $7.16 per thousand × 130 thousands = $930.80 monthly payment.

That is easy to do!

Unfortunately, it is not practical. In real life, anyone who gets a loan is told their monthly payment amount. No bank expects customers to figure it out themselves!

Knowing a Payment, Finding a Loan

We can also use the amortization table backwards to find how large a home someone can afford based on their desired monthly payment.

This is more practical for real life.

7. The Wahl family has an annual income of $45,000. They want to spend 20% of that income on a mortgage. How much are they prepared to pay each month?

7. The Wahl family wants to spend each year $45,000 × 0.2 = $9,000 per year.

This becomes $9,000 ÷ 12 = $750 per month.

8. Continuing with the Wahl family, if mortgage interest rates are at 6% then how large a twenty-year loan can they afford?

8. In the previous problem we found that the Wahl family could afford to spend $750 each month.

Imagine that the Wahl family goes to the bank with that $750.

The amortization table value for 6% and 20 years is $7.16. Every time the Wahl family gives the bank $7.16 they can get $1,000 more loan.

So $750 ÷ $7.16 ≈ 104.75.

They can give the bank that $7.16 amortization table amount 104.75 times before they run out of money.

Because each of those 104.75 instances of spending the amortization table amount got the Wahl family $1,000 more loan, our last step is to multiply by $1,000.

The loan size is 104.75 × $1,000 = $104,750 loan.

Often mortgage loans are rounded to the nearest thousand dollars. If you did that, then the answer would be a $105,000 loan.

9. Continuing with the Wahl family, how much will be paid total over the twenty years? How much of that is interest?

9. The monthly payment of $750 happens twelve times per year for twenty years.

$750 × 12 × 20 = $180,000 paid total.

Of that amount, all the money is either loan or interest.

So there is $180,000 − $104,750 = $75,250 interest total.

If you had earlier rounded the loan amount to $105,000 loan, then this problem would have an answer of $75,000 interest total.

Adding Real-Life Complications

A few real-life issues add to the cost of buying a home.

That is a lot of stuff! But it does not actually change the math very much.

We need to reduce the home value by the down payment to find the mortgage size. Then we can use the amortization table to find the monthly payment.

Those other details are usually paid up front by the buyer, but some might be paid by the seller or added onto the mortgage amount. We need to add up the items paid up front to find how much total cash the home buyer is required to have available.

10. A home is purchased for $150,000. The down payment is 20%. The mortgage has a 3% fee. The first month's mortgage payment must be paid up front at closing. Inspections cost $200. The various up front costs for insurance and fees total $950. The loan is for 30 years with a 5% rate. What is the mortgage amount? What is the mortgage monthly payment? How much cash must the buyer have available for all the up front costs?

10. The down payment is $150,000 × 0.2 = $30,000.

So the mortgage amount is $150,000 − $30,000 = $120,000.

The mortgage fee is $120,000 × 0.03 = $3,600.

The amortization table value for 5% and 30 years is $5.36. So the monthly payment is:

$5.36 per thousand × 120 thousands of loan = $643.20 monthly payment

The total up front costs are:

$30,000 mortgage + $3,600 mortgage fee + $643.20 first payment + $200 inspections + $950 other = $35,393.20

Formulas link to here link to index

Sum of Annuity Due

Sum of Annuity Due

One weakness of the simple and compound interest formulas was that the principal was the only deposit. Saving for retirement never works like that! Most households deposit money each year in a retirement account.

If a deposit is made each year the formula changes into the sum of annuity due formula.

Notice that the word "principal" is incorrectly used in the formula (for historic reasons) instead of "annual deposit".

The Sum of Annuity Due Formula (Version One)

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

Notice that when using this formula on a calculator you do not actually need nested parenthesis. The bits that look like (1 + rate) can just become numbers. For example, with a 5% rate we can just write 1.05. The bit that looks like (years + 1) can just become a number. For example, with a 40 year time span we can just write 41 in the exponent.

Some people like a shorter formula.

The Sum of Annuity Due Formula (Version Two)

Final Amount = Principal × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

Both versions of this formula might look big, but they are actually much easier to use than the compound interest formula. When using the compound interest formula we had to be alert for needing to divy up the annual interest rate into monthly or quarterly payouts. No longer! The sum of annuity due formula always involves annual deposits and payouts.

Also notice that the inside parenthesis can be avoided when using a calculator: if the rate was 6% we could just key in 1.06 in both places with (1 + rate) to skip the inner parenthesis.

So this formula might look scary, but it is actually less tricky than compound interest.

11. Someone saves $1,200 each year for retirement. If this is invested with 8% annual interest over 45 years, how much will it grow to be worth?

11. Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

= [ $1,200 × 1.0846 − $1,200 × 1.08] ÷ 0.08

≈ $500,911

Monthly Annuity?

A monthly version of the formula would also work. It would use monthly deposits, monthly interest, and (months + 1) as the exponent.

In our class we will only use an annual sum of annuity due formula. But perhaps a monthly version would help your household plan for retirement or be useful in a math project.

How Much to Save for Retirement

Typical Retirement

The general rule for retirement savings is to, at age 65, have saved $20 for each dollar that your retirement expenses will exceed your retirement income (from Social Security, pensions, etc.).

Once you retired, your savings would be gone in one year if you saved only $1 for each dollar that your expenses exceed your income. By saving 20 times that much, the interest your savings continue to earn should last for the rest of your life.

In other words, you have saved for the twenty years from age 65 to 84. The interest earned during those twenty years pays for the years of age 85 and beyond.

Typical American retirement expenses are greater than retirement income by about $20,000 per year. This means that for most Americans a good plan is to save $400,000 for retirement.

Cindy, Clara, and Chloe

three sisters retirement problem chart

Cindy, Clara, and Chloe are three sisters. Each has plans to save for retirement, but their plans are somewhat different.

Despite their different lives, by the time each was 25 years old she had the ability to set aside $3,000 per year for retirement. All three use a retirement account that earns 8% annual compound interest.

Which of them end up saving at least $400,000 for retirement?

12. Cindy annually puts the $3,000 into her retirement account for ten years (from age 25 to 34). Then she decides she has saved enough, and her kids are old enough to travel without hassle, so she switches to annually spending the $3,000 on a vacation for thirty years (from age 35 to 64). Her savings always grow at an annual rate of 8%. How much will Cindy have saved when she retires at age 65?

12. This is a two part problem. For the first part, we look at the first ten years. Regular deposits tell us to use the sum of annuity due formula.

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

= [ $3,000 × 1.0811 − $3,000 × 1.08] ÷ 0.08

≈ $46,936.46

For the second part, we look at the last thirty years. Merely watching an initial amount grow tells us to use the compound i interest formula.

Final Amount = Principal × (1 + interest rate per payout)number of payouts

= $46,936 × 1.0830

≈ $472,305.52

Cindy does save at least $400,000 for retirement. She also only had to save for ten years. By starting early, she benefitted from the tremendous power of earning interest over a long time period.

13. Clara wants to enjoy herself before she gets older, and annually spends the $3,000 on a vacation (from age 25 to 34). Then she starts worrying about retirement, so she switches to annually putting the $3,000 into her retirement account for thirty years (from age 35 to 64). Her savings always grow at an annual rate of 8%. How much will Clara have saved when she retires at age 65?

13. This is not a two part problem. For the first ten years no savings happen! The only math is for the last thirty years. Regular deposits tell us to use the sum of annuity due formula.

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

= [ $3,000 × 1.0831 − $3,000 × 1.08] ÷ 0.08

≈ $367,037.60

Clara almost saves at least $400,000 for retirement, but fails. That happened even though she put a lot of money into savings for thirty years! By starting late, she missed out on much of the benefit of earning interest over a long time period.

14. Chloe likes consistency. Every year she puts $1,500 into her retirement account, and spends $1,500 on a vacation (from age 25 to 64). Her savings always grow at an annual rate of 8%. How much will Chloe have saved when she retires at age 65?

14. This is not a two part problem. For all forty years regular deposits tell us to use the sum of annuity due formula.

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

= [ $1,500 × 1.0841 − $1,500 × 1.08] ÷ 0.08

≈ $419,671.56

Chloe does save at least $400,000 for retirement. (Although not as much as Clara.) She did start early enough to benefit from the power of earning interest over a long time period.

In real life, most households are a mix of Cindy and Chloe. They do not save anything in early years. Then they save some money but not enough.

For many households, the best way to save more is to kick a bad habit. The next problem uses cigarettes as an example of an expense that could be changed into savings. Perhaps for your household, the problem should instead discuss eating at restaurants too often, or buying fancy morning coffees?

cigarette money

15. Cliff is Clara's brother. He stops smoking at age 25, and decides to devote the money he used to spend on cigarettes to retirement. He used to smoke 1 pack per day, at $5.70 per pack. How much money per year was Cliff spending on cigarettes? If he instead puts his "cigarette money" annually into a retirement account that earns 6% annual compound interest, how much will extra will he have for retirement after forty years?

15. The annual deposit is $5.70 × 365 = $2,080.50

For all forty years regular deposits tell us to use the sum of annuity due formula.

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

= [ $2,080.50 × 1.0641 − $2,080.50 × 1.06] ÷ 0.06

≈ $341,301.21

Cliff almost saves the $400,000 his household wants for retirement simply by changing his cigarette money into savings!

When you did the math for Clara, did you think to yourself, "Bah! Who has $3,000 per year to put into savings at age 25? This is unrealistic!" That complaint does make sense.

But Cliff shows that sometimes a lifestyle change is sufficient to claim the power of starting to save early. Every bit counts, even if not as much as Clara's $3,000 per year! Cliff and his household barely need to do any other savings to be ready for retirement.

Homework link to here link to index

Not So Fast, My Friend

Go back and actively read these online lecture notes a second time.

• Each time you get to an example problem, try to solve the problem yourself before looking at the solution.

• If you get stuck, only look at the solution one line at a time. Either scroll down very carefully or hold a piece of paper over the screen to block the lower text.

• If you are still stuck, contact the instructor or bring questions to class!

Remember the three levels of math understanding. Reviewing the notes actively can work as prompt homework problems where the goal is merely to find out how much time you will need for studying and how to pace yourself—not yet to finish problems and master topics.

More Practice

16. Remember the Wahl family in problems #7-9? They budgeted $750 per month on a mortgage, found a $104,750 loan, over twenty years paid $180,000, and of that $75,250 was interest. Does doubling the monthly payment double the size of the other numbers? Consider the Moneybag family that has spends $1,500 per month on a mortgage, with everything else the same as before (20 years, 6% rate). How large a loan can they afford? How much will be paid total over the twenty years? How much of that is interest?

16. Yes, doubling the amount you pay per month makes all the other numbers double.

The Moneybag family pays $1,500 per month for a $209,500 loan. Over the twenty years they pay $360,000 total, of which $150,500 is interest.

As with problems #8-9, you could decide that mortgage loans are often rounded to the nearest thousand dollars and so change your answers to a $210,000 loan with total interest paid of $150,000.

17. Does halving the number of years either halve or double the size of the numbers? Next consider Mr. Short, who wants a ten-year loan, with everything else the same as before ($750 monthly payment, 6% rate). How large a loan can he afford? How much will be paid total over the ten years? How much of that is interest?

17. Some number halve. Others do not.

Mr. Short still has a monthly payment of $750. Halving the years to ten years changes the amortization table value to $11.10. The new mortage size is $750 ÷ $11.10 ≈ 68, so he gets a $68,000 loan.

No, halving the number of years did not halve the morgage size. The amortization table does not work that way.

The monthly payment is unchanged, so $750 happens twelve times per year but only for ten years. $750 × 12 × 10 = $90,000 paid total

Yes, the total paid over half as many years will be half of what was paid before.

The total interest paid is $90,000 − $68,000 = $22,000 interest total

No, the total interest paid is not half of before. It is much less than half! Shorter loans save a lot of money!

18. Brenda can afford to spend $900 per month on mortgage payments. Currently mortgage rates are 8% per year. How big a twenty-year mortgage can she afford? How big a thirty-year mortgage can she afford?

18. The amortization table value for 8% and 20 years is $8.36, which results in a $107,700 loan. The amortization table value for 8% and 30 years is $7.34, which results in a $122,600 loan. (You could round these estimates differently.)

As before, you could decide that mortgage loans are often rounded to the nearest thousand dollars and so change your answers to a $108,000 twenty-year loan and a $123,000 thirty-year loan.

market increase image

19. One investment company loans the Moneybag family that $209,500 loan, earning $150,500 interest over twenty years (from problem #16). A rival investment company invests in index stocks that appreciate as much as the overall stock market. The historical return for the stock market is about 11%. How much more does the rival make? Why would the first investment company choose to offer a mortgage considering that the rivals made so much more?

19. Merely watching an initial amount grow tells us to use the compound interest formula.

Final Amount = Principal × (1 + interest rate per payout)number of payouts

= $209,500 × 1.1120

≈ $1,689,054.27

Subtract away the initial $209,500 to find a net gain of $1,479,554.27

The rival investment company makes $1,479,554.27 − $150,500 = $1,329,054.27 more! Yowzers!

The first investment company chooses to offer a mortgage because they do not want risk. If they were willing to have any risk—even a bond paying a measley 3% annual interest—they could earn more money than by offering that mortgage.

Random Homework Problems

Homework

Each time you load the page these problems change!

Random Review

Math 20 Topics

Here are Math 20 review problems about unit analysis. Each time you load the page these problems change!

Icing on the Cake

Icing on a Gingerbread House

27. Zane has an annual income of $65,000. He wants to spend 30% of his income on a mortgage, with a 20-year loan. The interest rate is 5%. How large a loan can he afford?

28. How much will Zane pay total over the 20 years?

29. How much of what Zane pays over the 20 years is interest?

30. Maria saved for retirement for 31 years, by depositing $1,600 each year into an accout with 8% annual interest. How much was in the account at the end?

31. Mario saved for retirement for 45 years, by depositing $980 each year into an accout with 8% annual interest. How much was in the account at the end?

32. According to research by Vanguard, the average 25-year-old American has saved $3,865 for retirement. If this single deposit is invested with 9% compound interest for 40 years, how much will it grow to be worth?

average savings chart

33. That same chart says the average 65-year-old American has saved $176,696. How much more is this than your answer to the previous problem? (In other words, how much more does the average American save, beyond their nest egg at age 25?)

34. That same chart says the average American who retires at age 65 has saved about $176,696 for retirement. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to that amount, over 40 years and at 9% annual interest. (Hint: an annual deposit of $400 is too small, but $500 is too big.)

Problems #32, #33, and #34 show us that it is easy to save as much as an average American. Less than $500 per year is enough to be average. Do not be embarassed if you start small with retirement savings. Become above-average by saving more later in life as your career progresses.

Concept Tasks

cover image from Aha Insight book by Martin Gardener

These homework problems test your understanding in a deeper way. You may turn in written work or choose to share your work in front of the class. Be prepared to defend your understanding when your instructor asks you another question or two about your work.

35. When the length of the term (years) of a loan is made longer, does the monthly payment increase or decrease? Does the total amount paid increase or decrease?

36. When the length of the term (years) of a loan is made longer, does the affordable mortgage size increase or decrease?

37. Remember that general rule for retirement that advised saving $400,000 for retirement, to be able to spend $20,000 per year when retirement expenses exceed retirement income? A similar but stricter rule for retirement savings advises spending 3.5% of your retirement savings each year while retired. If a household did save $400,000 for retirement, how much could they spend each year according to this stricter rule? If they wanted to spend $20,000 per year, how much does the stricter rule advise them to save before retiring?

38. How does the long-term financial cost of infant care compare to saving for kids' college expenses? The "big three" child care centers in Eugene cost roughly $13,000 per year. (Vivian Olum at UO, Early Learning Childrens Community at LCC, and Child Development Center at EWEB—Oregon is an unusually expensive state for child care.) Use the sum of annuity due formula for two years (when the infant is age 1 and 2), and then the compound interest formula for sixteen years (until the child is 18). If the family insted saved the infant care money and earned a 5% annual interest rate, how much would be saved towards the child's college tuition?

Archer Queen from Clash of Clans

39. In a certain computer game the archer queen gets 10% stronger every time she gains a level. Two brothers play the game. The younger brother has a level 20 archer queen. The older brother has a level 27 archer queen. Roughly how much stronger is the older brother's archer queen?

40. A girl who was saving her money in a piggy bank to purchase a ukulele instead received one for her birthday. So she decided to spend 4% of her remaining piggy bank savings every week on other stuff. About how many weeks will this happen, before she spends half of those savings?

This last problem has no absolutely correct answer, even though some people might have strong opinions about why an answer is right for them. The graph is strange because a single person appears multiple times, hidden in different places.

41. Gina's mother keeps pestering her to "shop for a husband while you are young and pretty." Gina's friends assure her that this advice is outdated, and Gina can focus on her career at least until the age of 30 if not 35. Use the graph of census data to decide who is correct. (Hint: a woman who is 35 years old in 2014 appears in four different places on this graph!)

2014 Never Married Census Data graphed by Dalrock

LCC Math 25 Packet Problems

Percent review problems are in the packet on pages:
  • PF-1 to PF-5
  • MS-28 to MS-29

Charge options problems are in the packet on pages:
  • PF-42 to PF-44

Mortgage problems are in the packet on pages:
  • PF-31 to PF-36

Sum of annuity due problems are in the packet on pages:
  • PF-65 to PF-73
  • PF-60 to PF-64