Real Estate Math Problems
Steve bought a piece of property for 25% less than the listed price. He later sold it for the listed price. His gain was:
If the listing price is $100,000, 25% less would be $75,000. Later selling at $100,000 would be a gain of $25,000 divided by $75,000 = .33 (33%).
Jane purchased a house for $100,000 with a down payment of 25% of the purchase price. The balance was financed on a 30-year amortized loan with interest at 8% per annum. The lender requires monthly impounds for property taxes of $1,000 per year and casualty insurance costing $788 for a three year policy. Assuming that the first monthly payment on the principal is $217, the total amount the buyer will have to pay the first month will be approximately:
$100,000 x 25% = $25,000 cash down payment. $100,000 - $25,000 = $75,000 loan amount. $75,000 x 8% = $6,000 annual interest. $6,000 divided by 12 = $500 monthly interest. $788 divided by 36 = $21.89 monthly insurance. $1,000 divided by 12 = $83.33 monthly taxes. $500 + $21.89 + $83.33 + $217 principal = $822.22 total first month payment.
Ned just sold his house for 9% less than he paid for it. The selling price was $105,900. What was the original purchase price?
$105,900 represents 91% of the original price. $105,900 divided by 91% = $116,373.63(original purchase price).
A property sold for $110,000. Buyer Tom financed 90%. Approximately how much will the monthly PMI increase his payment if the renewal rate is .30%?
$110,000 x 90% = $99,000 (amount of loan) x .30% = $297 yearly renewal rate. 297 divided by 12 = 24.75 monthly increase.
A buyer purchased a home for $125,000 putting up a good down payment of 20% of the purchase price, and financing the balance on a 25-year amortized loan with interest at 8.25% per annum. The bank requires monthly impounds for property taxes of $750 per year and casualty insurance costing $945 for a three year policy. Assuming that the first monthly payment on the principal is $155, the total amount the buyer will have to pay the first month will be approximately:
$125,000 x 20% = $25,000 cash down payment. $125,000 - $25,000 = $100,000 loan amount. $100,000 x 8.25% = $8,250 annual interest. $8.250 divided by 12 = $687.50 monthly interest. $945 divided by 36 = $26.25 monthly insurance. $750 divided by 12 = $62.50 monthly taxes. $687.50 + $26.25 + $62.50 + $155 principal = $931.25 total first month payment.
A property is sold for $125,000. If the lender charges a 2% loan fee for an 80% loan, what is the amount of the fee?
$125,000 x 80% = $100,000 amount of loan. 100,000 x 2% loan fee = 2,000.
Frank just sold his house for 6% more than he paid for it. The selling price was $140,450. What was the original purchase price?
$140,450 divided by 106% = $132,500.
Jay is the beneficiary on a trust deed. The annual interest rate is 8%. If in 5 years he has received $4,500 in interest, what is the principal amount of the loan?
$4,500 divided by 5 years = $900 interest per year. $900 divided by 8% = $11,250 principal.
A homeowner has a balance of $149,570.75 remaining on the mortgage. The interest rate is 9.5% and the monthly payment is $1,303.55. After the next two payments, the balance will be:
$149,570.75 x 0.095 ~ 12 = $1,184.10 month interest; $1,303.55- $1,184.10 = $119.45 principal; $149,570.75 -$119.45 = $149,451.30; $149,451.30 x 0.095 ~ 12 = $1,183.16 month interest; $1,303.55$1,183.16 = $120.39 principal; $149,451.30- $120.39 = $149,330.91.
Ken purchased a property for $16,380. If he wants to sell it without a loss for a price which would include a 9% commission, the value of the property would have to appreciate:
$16,380 divided by .91 = $18,000. $18,000 - $16,380 = $1,620.
An owner of a home listed it at a price which would leave him $16,800 after the broker received a 6% commission. If the broker sold the property at the listed price, how much commission would he receive?
$16,800 divided by 94% = $17,872 - $16,800 = $1,072 commission.
Ms. Williams purchased a house in a run-down condition and spent an amount equal to 10% of the purchase to fix it up and bring it to its present value. The house in now worth $18,700. The cost of the house is equal to the existing loan of the property. The amount of the loan is:
$18,700 = 100% purchase price + 10% fix-up money. $18,700 divided by 110% = $17,000.
The investor criterion for a home mortgage is an uninsured loan-to-value ratio of 90% of the appraisal. The sales agreement and appraisal is in the amount of $180,000. Following underwriting guidelines, the buyer qualifies for a loan of $145,000. How much of the purchase will be financed by this investor?
$180,000 x 0.9 = $162,000 maximum uninsured loan, so the buyer's entire loan of $145,000 can be figured by the described investor.
Broker John and Broker Jay agreed to divide a 4 1/2% commission equally on the sale of a home, which sold for $180,000. Sue, the listing salesperson, works for Broker John on a 50-50 commission split. Sue would receive approximately how much commission as a result of the sale?
$180,000 x 4 1/2% = $8,100 total commission. $8,100 divided by 2 = $4,050 commission to Broker John's office. $4,050 divided by 2 = $2,025 commission to Sue.
Bob owns a home, which is free and clear. He recently purchased a new home. The contract to purchase the new home however, has a contingency that Bob's current home be sold within 60 days and that Bob net $19,500 from the sale after all expenses have been deducted. Bob employs broker Jones to assist him in the sale of the property. The broker is to receive a 6% commission and there will be an escrow fee of $300. How much must the home in Irvine sell for in order to net Bob $19,500?
$19,500 cash (net to owner after expenses) + 300 cash (to cover escrow fee) = $19,800 total cash needed except for the cash needed to pay the 6% commission. Therefore $19,800 represents 94% of what the selling price should be. $19,800 divided by 94% = $21,063.
An apartment's financial statement disclosed the following: Gross income = $20,000; vacancy factor = 9%; operating expenses = $5,000; payments to amortize the existing 1st and 2nd deeds of trust = $300 monthly. Using a 12% capitalization rate, what is the value of the property?
$20,000 (gross income) - $1,800 (vacancy factor) = $18,200 (effective gross income). $18,200 - $5,000 (operating expenses) = $13,200 (net income). $13,200 divided by .12 (cap. rate) = $110,000. Note: You cannot subtract principal or interest payments on trust deeds. This is not an operating expense.
The net income of an apartment building went down $200 per month when a freeway was built nearby. If investors demand a 10% capitalization rate for this area, how much has the building lost in value?
$200 x 12 = $2,400. $2,400 divided by 10% = $24,000.
How many acres are in the NW 1/4 of the SE 1/4 of Section 10 and the SE 1/4 of Section 3 and the N 1/2 of the NE 1/4 of Section 16?
How many acres are in the NW 1/4 of the SE 1/4 of Section 10 and the SE 1/4 of Section 3 and the N 1/2 of the NE 1/4 of Section 16?
Brad bought a parcel of land in 1989 and subdivided it into four separate lots. Five years later, he sold each lot for $75,000. The adjusted basis for each lot was $25,000. Brad's capital gain on these transactions is:
$25,000 x 4 lots = $100,000 adjusted basis. $75,000 x 4 lots = $300,000 total selling price. $300,000 - $100,000 = $200,000 capital gain.
In order to earn $250 per month, the amount an investor would have to invest at 6% is:
$250 income x 12 months = $3,000 annual income. $3,000 annual income divided by 6% rate of return = $50,000 total required investment.
Lynn sold a home for $27,000. She made a 15% profit. What did the home cost?
$27,000 divided by 115% = $23,478.
An apartment building is valued at $275,000 using a 7% capitalization rate. How much would an investor pay for the property if she demanded a 9% capitalization rate?
$275,000 x 7% = $19,250 annual net income. $19,250 divided by 9% = $213,889.89. Remember, as capitalization rate goes up, the value goes down.
Investors want to build an apartment house which will have a gross income of $3,000 per month and annual expenses of $7,000. The investors require a 8% capitalization rate on the investment. If the improvements cost $175,000, how much can the investors pay for the land?
$3,000 x 12 = $36,000 gross annual income. $36,000 - $7,000 (expenses) = $29,000 annual net income. $29,000 divided by 8% (cap rate) = $362,500 total property value (land and improvements). $362,500 - $175,000 (cost of improvements) = $187,500 available to pay for the land.
Ken, an investor, wants to build a rental property which will have a gross income of $3,200 per month and annual expenses of $6,500. The investors require a 10% capitalization rate on the investment. If the improvements cost $200,000, how much can the investors pay for the land?
$3,200 x 12 = $38,400 gross annual income. $38,400 - $6,500 (expenses) = $31,900 annual net income. $31,900 divided by 10% (cap rate) = $319,000 total property value (land and improvements). $319,000 - $200,000 = $119,000 available to pay for the land.
What is the salesperson's 45% of a 6% commission on a $31,000 sale?
$31,000 x 0.06 = $1,860 x 0.45 = $837.
Ted borrowed $40,000 from a bank to purchase a lot. He paid 4 points to get the loan. The loan contained a 4% prepayment penalty. His monthly payment was $160.00, including interest at 8% per year. Five years later, John sold this real estate and paid-off the loan. If the loan had an average loan balance of $38,500 during the five years Ted owned the property, what was the bank's gross profit on this loan?
$40,000 x 4% = $1,600 points. $40,000 x 4% = $1,600 prepayment penalty. $38,500 x 8% = $3,080 interest per year. $3,080 x 5 years = $15,400. $1,600 + $1,600 + $15,400 = $18,600 gross profit by bank.
A seller has $5,000 closing costs, a $93,000 loan balance, and pays 7% commission on a $125,000 sale. What are his net proceeds from the sale?
$5,000 + $93,000 = $98,000 total dollar expenses of closing. Convert 7% to a decimal and multiply. $125,000 x .07 = $8,750 commission; $125,000— $98,000— $8,750 = $18,250 net proceeds to seller.
$5,000 investment at 10% interest compounded annually earns how much in three years?
$5,000 x 10% = $500 (interest for 1st year). $5,000 + $500 = $5,500 x 10% = $550 (interest for the 2nd year). $5,500 + $550 = $6,050 x 10% = $605 (interest for the 3rd year). $500 + $550 + $605 = $1,655 interest earned in 3 years. Remember, compound interest is interest on the principal and prior interest earned
A $5,000 investment at 10% interest compounded annually earns how much in three years?
$5,000 x 10% = $500 (interest for 1st year). $5,000 + $500 = $5,500 x 10% = $550 (interest for the 2nd year). $5,500 + $550 = $6,050 x 10% = $605 (interest for the 3rd year). $500 + $550 + $605 = $1,655 interest earned in 3 years. Remember, compound interest is interest on the principal and prior interest earned.
Mary sold her house and took back a second trust deed for $5,400. She immediately sold it for $4,050. What was the rate of discount?
$5,400 - $4,050 = $1,350 (amount discounted) divided by $5,400 = .25 (25%).
Ted paid $85,000 for a house, putting up a 15% cash down payment and borrowing the balance at 8.5% interest. The loan is payable over 30 years with monthly payments of $560, including principal and interest. If Ted makes the payments as scheduled, the percentage that the original cost of the house increased because of the use of credit is?
$560 x 360 months = $201,600 (loan payments). $85,000 x 15% = $12,750 (down payment). $201,600 + $12,750 = $214,350 (total cost). $214,350 divided by $85,000 x 100 = 252.2%. 252.2% - 100% (original cost) = 152.2% increase.
A property sold for $57,300 and is subject to revenue stamp fees of .50 for each $500 (or portion thereof) of the sales price. What is the cost of the stamps?
$57,300 — $500 = $114.60 souse $115 (because the problem states "any portion thereof"); $115 x .50 = $57.50.
Steve purchased a note worth $6,000 for $5,500. Interest rate on the note is 5%. What was his percentage of profit made on his investment if the note is paid off in one year?
$6,000 x .05 = $300. $6,000 - $5,500 = $500. $300 + $500 = $800 divided by $5,500 = .145 (14.5%).
How much interest would a $6,000 loan earn in five years at 10% simple interest?
$6,000 x 10% = $600 per year x 5 years = $3,000.
A loan was made for $15,000 at an interest rate of 5% per year for a 20-year period. The total interest that would be paid over the entire loan period if the monthly payment for this fully amortized loan is $6.50 per $1,000, would be approximately:
$6.50 per $1,000 x 15 (thousand) = $97.50 monthly payment. 20 years x 12 months = 240 monthly payments. $97.50 x 240 = $23,400.00. $23,400.00 - $15,000 (money borrowed) = $8,400.00 total interest paid.
Tom obtains an interest-only purchase money loan for $60,000. The interest rate is 9.5% and the term is 30 years. How much interest will Tom have paid at the end of the 30-year term?
$60,000 x 9.5% = $5,700 interest per year. $5,700 x 30 years = $171,000 total interest.
Jerry spends 25% of her gross monthly income on her apartment rent of $600. What is her gross monthly income?
$600 divided by 25% = $2,400.
A small income property generates a monthly gross income of $600. Over the last five years it has been vacant for three months. The annual expenses are $2,000. If an appraiser applied a 10% capitalization rate to this property, what would be the value?
$600 x 12 = $7,200 annual gross income. 12 months x 5 years = 60 months. 3 months vacant divided by 60 months = .05 (5%) vacancy factor. $7,200 x 5% = $360 annual vacancy loss. $7,200 - $360 = $6,840 adjusted gross income. $6,840 - $2,000 annual expenses = $4,840 annual net income. $4,840 divided by 10% = $48,400.
A $7,000 loan is sold to an investor for $6,500. It is a straight note, due and payable at the end of one year. It bears a 9% interest rate. What percentage return on the principal dollar invested will be made by the investor?
$7,000 x 9% = $630 interest paid on the loan. $7,000 - $6,500 = $500 made by investor as a result of the discount. $630 + $500 = $1,130 total profit made by investor. $1,130 divided by $6,500 = 17.38%.
A buyer is obtaining a conventional loan that requires 29/33 ratios. He earns $75,000 a year, and has a $450 car payment. What is his maximum PITI payment?
$75,000 + 12 = $6,250 monthly income. $6,250 x .29 = $1812.50 front end qualifier. $6,250 x .33 = 2062.50— $450 debt = $1612.50 back end qualifier. The maximum PITI (Principal, Interest, Taxes, and Insurance) is the lower of these two qualifiers.
Inventory in a store costs $8,500 net. It was sold for 20% profit. However, 10% of the gross profit was lost due to bad credit risks over a period of one year. How much profit did the store owner make that year?
$8,500 x .20 = $1,700 x .10 = $170. $1,700 - $170 = $1,530.
A 20-unit apartment building is valued at $800,000 based on a 10% capitalization rate. If the rent for each unit increased $20 per month, what would the value of the same building be based on a capitalization rate of 16%?
$800,000 x .10 = $80,000. 20 units x $20 = $400 per month. $400 x 12 = $4,800 per year. $80,000 + $4,800 = $84,800. $84,800 divided by .16 = $530,000.
Jim purchased a home for $85,000, with a $25,000 down payment and financed the balance with a $60,000 straight note. He then sold the property for double the purchase price, paying no principal or interest payments on the loan during his ownership. Each dollar invested is now worth?
$85,000 x 2 = $170,000 new selling price. $170,000 - $60,000 loan = $110,000 profit. $110,000 (profit) divided by $25,000 (invested) = $4.40.
John bought a house four years ago for $80,000. He just sold it for $86,900. What was the percent of profit?
$86,900 - $80,000 = $6,900 (profit) divided by $80,000 = .086 (8.6%).
A man paid $9,500 for the stock in trade of a retail business. He sold it for 25% more than he paid for it. If he lost 15% of the gross sale price because of credit risks, the net profit on his original investment would be:
$9,500 x 25% = $2,375. $9,500 + $2,375 = $11,875 x .15 = $1,781.25. $11,875 - $1,781.25 = $10,093.75 - $9,500 = $593.75.
The Bakers bought a vacation home with a $90,000 first mortgage. Current conventional rates are 10.5%, 30-year term. The seller agreed to pay three points, which will amount to:
$90,000 x 3% = $2,700.
In order to be able to earn $90.00 per month from an investment that yields a 6% return, you must invest?
$90.00 per month x 12 = $1,080 per year. $1,080 divided by .06 = $18,000. Remember to work the problem using the annual income and the annual interest rate.
Steve sold his home which had no loans against it. He received a settlement check from escrow for the amount of $96,400 after paying escrow fees of $1,365.00 plus a 6% real estate broker's commission. What was the selling price?
$96,400 + $1,365.80 = $97,765.00 net amount after paying commission. $97,765.00 divided by 94% = $104,005.32 selling price.
The lender's underwriting criteria would accept a housing expense to income ratio of 33% and a ratio of total debt service of up to a maximum of 38% of monthly gross income. The applicant has an outstanding college loan payable at the rate of $150.00 per month and a car payment of $450.00 monthly. Gross annual income is established at $100,000 per annum. What is the maximum PITT the lender will approve?
($100,000/12) x 0.38 is $3,166.67 - ($150 + $450) = $2,566.67
A lender loans borrower Smith a sum of $1,800 and takes back a note and deed of trust. The note was set up to be fully amortized over a 1-year period, interest at 8% per year, payable at $162 per month. When the note was drawn, it was immediately sold to an investor for a 15% discount. If the loan were fully paid off at the end of the year, what percent return would the investor make on this investment?
12 monthly payments x $162 per month (includes principal and interest) = $1,944 total amount received. $1,800 x 15% (discount) = $270. $1,800 - $270 = $1,530 (cost to investor). $1,944 (amount received) - $1,530 (cost to investor) = $414 profit made. $414 divided by $1,530 x 100 = 27% return.
Assume it would cost $16,500 to reproduce an improvement today. The land is valued at $2,000 today. Assuming the improvement is actually 12 years old and the depreciation rate is 2-1/2%, what is the depreciated value today?
12 years x 2-1/2% = 30% depreciation rate. $16,500 x 30% = $4,950 depreciation. $16,500 - $4,950 = $11,550 remaining value. $11,550 + $2,000 land value = $13,550.
A house was purchased 5 years ago for $90,000. The area has generally appreciated at an annual rate of 2%. How much should the house sell for now?
5 years 2% per year = 10% appreciation (straight line); 100% + 10% = 110% property is now worth relative to original purchase price; $90,000 x 110% = $99,000 current value.
If a lender agrees to make a loan based on an 80% LTV, what is the amount of a loan for a property appraised for $135,000 and a sale price of $137,800?
80% LTV is based on the lesser of the appraised value or the sale price; $135,000 x .80 = $108,000.
Rita Morgan has $86,576 left on her 8.5% mortgage. Her monthly payment is set at $852.56 for principal and interest (she pays her own taxes and insurance). How much of her next payment will go to reduce the principal?
A year's interest on the present debt would be $7,358.96 ($86,576 x 0.085). A month's interest is $613.25 ($7,358.96 ~ 12). The principal portion of her payment is $239.31 ($852.56 —$613.25).
If the buyer's agent splits his commission 50/50 with his broker, and the seller's agent receives 60% of his broker's 50% share, how much does the buyer's broker receive if the listing broker's fee is 6% of the $125,000 sales price of the house?
Convert 6% to a decimal and multiply; $125,000 x .06 = $7,500 total commission on sales price. Convert 50% to a decimal and multiply; $7,500 x .50 = $3,750 total buyer's broker share.
What is 75% of $250,000?
Convert 75% to a decimal and multiply; $250,000 x.75 = $187,500.
Mrs. Adams owned an income property with an adjusted cost basis of $160,000 and a fair market value of $240,000. She exchanged the property for another income property which had a fair market value of $230,000. Both properties had no loans against them and no adjustment was made for the differences in value. For federal income tax purposes, the new property will have a basis for Mrs. Johnson of:
In a tax deferred exchange, the cost basis of the property being traded becomes the cost basis for the property being acquired if no adjustment is made for differences in value; in this case, $160,000.
Frank purchased a home for $80,000. He financed the purchase with a 90% conventional fixed-rate loan. The loan fee charged by the lender was 2%. The private mortgage insurance company charged a .50% first-year premium and a .75% renewal premium. What was the loan fee? The first-year premium? The monthly renewal premium?
Loan fee = $80,000 x 90% x 2% = $1,440. First year premium = $80,000 x 90% x .5% = $360. Monthly renewal = $80,000 x 90% x .75% divided by 12 = $45.
To receive a 12% return on an investment of $750,000, what would be the required net operating income?
Net income = purchase price x capitalization rate. $750,000 x 12% = $90,000.
Mr. Jones, an owner of a packaging firm, purchased a new machine in 1991 and paid $5,500. It was estimated at the time of the purchase to have a total economic life of 10 years and a salvage value of $550. Using the straight line method of depreciation, the book value at the end of 7 years would be:
Salvage value must be deducted before computing depreciation. $5,500 - $550 = $4,950 divided by 10 = $495 per year x 7 years = $3,465. $5,500 - $3,465 = $2,035 (book value).
The Hatchers have a gross income of $60,000. The lender wants them to spend no more than 28% of their income on their housing expenses. A house they can buy for $240,000 has $2,400 in annual property taxes. Homeowners insurance would cost about $400 a year. At today's interest rates, monthly payments would be $6.65 per $1,000 borrowed on a 30-year mortgage. What is the smallest amount they can expect to spend for a cash down payment?
The Hatchers' monthly gross income is $5,000 ($60,000 ~ 12). The lender allows them a monthly mortgage payment no higher than 28% of $5,000 = $1,400. From that $1,400, a month's property taxes must be subtracted ($2,400 ~ 12 = $200.00) and a month's homeowner insurance premium ($400 ~ 12 = $33.33). That leaves $1,167 to pay for the mortgage itself, principal, and interest. At $6.65 per thousand dollars, $1,167 will pay for $175.49, or $175,490 borrowed on a mortgage ($1,167 ~ $6.65= $175.49). If they buy for $240,000 and borrow $175,490, the Hatchers will need $64,510 as a cash down payment ($240,000 - $175,490).
'FHA mortgage insurance premium is calculated at a rate of 0.5% annually. How much is the premium for the month in which the remaining principal owed is $92,347?
The annual premium on $92,347 would be $461.74 ($92,347 x 0.005). One month's premium is $38.48 ($461.74/12).
An individual borrowed $9,000 on a straight note and an interest of 7%. If the total interest paid on the note totaled $840.00, the term of the loan was:
To determine this, you multiply $9,000 times 7%, which equals $630.00 interest per year. Next $630.00 divided by 12 equals $52.50 per month. And finally $840.00 total interest divided by $52.50 equals 16 months.
Bill sold his office building for $185,000. This was 15% more than what he paid for it. His original cost was approximately
To find 115%of the purchase price, divide $185,000 by 1.15 to produce an answer of $160,869.
What would the transfer tax be on the sale of a lot that sold for $10,500 with a loan assumed at $10,400?
Transfer tax is paid on the difference between the selling price and any existing trust deeds that are assumed by the buyer in excess of $100. The first $100 is exempt.
The Smiths offer to buy the Brown's house for $125,000, with the closing scheduled for June 15. The Browns sign a written acceptance, with the provision that the closing is to be on June 16. At this point, which of the following can occur?
When the Browns made a counteroffer, it constituted a rejection of the original offer. The Smiths are now free and may accept or reject the counteroffer.
Under Article 7, if the actual costs are $159 on a $4,000 loan, the maximum costs that can be charged is:
The broker charges the actual costs or the maximum costs, whichever is less. The maximum costs for this loan are $390, therefore the broker may only charge the actual costs of $159.
A lender is providing 90% of the financing for a new house. If the house appraises for $75,000, what is the buyer's down payment?
The down payment is the difference between the sales price and the loan amount. 100% - 90% = 10% down payment. Convert 10% to a decimal and multiply. $75,000 x .10 = $7,500
Sue sold her property for $27,500 and realized a 7% profit over what she paid for it. Her profit was:
The selling price of $27,500 = cost + 7% or 107% of selling price. $27,500 divided by 107% = $25,700 purchase price. $27,500 - $25,700 = $1,800 profit.
