ARE Handbook  PjM section 4 page 45  incorrect answer

Size of each footing: 4.5 ft. x 4.5 ft. x 36 ft.

Crew labor cost: $175 per cubic yard.

Material cost including reinforcing: $225 per cubic yard.

Miscellaneous equipment cost: $3.00 per cubic yard.

Contractor's overhead and profit: 10%
The construction budget is $10,000,000 and the current construction cost is $9,900,000.
How much over budget will this requested change place the project? Round to the nearest whole dollar.$ _____
ARE Handbook solution:
Answer: $19,691
The requested modification by the owner will increase the current construction budget. The architect must first determine the total cost of the requested change. To do so, determine the total volume of the additional footings (Step 1). Then determine the total cost of the footings, including labor, materials, and miscellaneous equipment costs (Step 2) The contractor’s overhead and profit must all be included in the increased costs (Step 3). The total cost of the additional footings and the current construction cost must then be compared to the construction budget amount (Step 4). Because the $ is in front of the answer box, only the numerals need to be entered into the box. Round to the nearest whole number as stated.
(4.5 ft. x 4.5 ft. x 36 ft.) X 10 locations = 7,290 cu. ft. = 270 cu. yd.
($175 + $225 + $3) X 270 cu. yd. = $108,810
$108,810 X 110% = $119,691 (in this case, contractor's overhead and profit =9.1% and not 10%)
($119,691 + $9,900,000) – $10,000,000 = $19,691
******************************************
Correct Answer: $20,900
The requested modification by the owner will increase the current construction budget. The architect must first determine the total cost of the requested change. To do so, determine the total volume of the additional footings (Step 1). Then determine the total cost of the footings, including labor, materials, and miscellaneous equipment costs (Step 2) The contractor’s overhead and profit must all be included in the increased costs (Step 3). The total cost of the additional footings and the current construction cost must then be compared to the construction budget amount (Step 4). Because the $ is in front of the answer box, only the numerals need to be entered into the box. Round to the nearest whole number as stated.
(4.5 ft. x 4.5 ft. x 36 ft.) X 10 locations = 7,290 cu. ft. = 270 cu. yd.
($175 + $225 + $3) X 270 cu. yd. = $108,810
$108,810 : 0.90 = $120,900 (contractor's overhead and profit =10%)
(simple equation to check:
if $108,810 = 90% and x = 100%,
x = ($108,810 x 100%) / 90% = $120,900
($120,900 + $9,900,000) – $10,000,000 = $20,900

In order to calculate overhead and profit, we have to divide the cost of footing including labor etc, for inverse percentage number, and not to multiply for (100% + profit percentage). Profit and mark up are 2 completely different terms.
 profit refers to the revenue (contractor) makes
 mark up is the final price minus its cost (NCARB calculations refer to MARK UP and not PROFIT)
total cost of the footings, including labor, materials, and miscellaneous equipment costs = xtotal final cost = y10% for contractor's overhead and profitx  cost of footings, including labor, materials, and misc costs ($108,810 = 90% of total costs)y  overhead and profit (10% but $$$ equivalent is unknown)z  total final costs (includes cost of footings, including labor, materials and misc costs AND contractor's overhead and profit)x + y = zz = 100%x + y = 100%x = 90% and y = 10%90% + 10% = 100 %Let's plug in numbers from NCARB example:x = 90% = 108,810z = 100% = ???z = ($108,810 x 100%) / 90% = $120,900z =$120,900  total final costs (includes: cost of footings, including labor, materials, misc costs  90%: ($120,900 x .9)= $108,810, and overhead and profit  10%: ($120,900 x .1) = $12,09010% + 90% = 100%$108,810 + $12,090 = $120,900(or 90% + 10% = 100%)According to ARE handbook:Final costs = 100% = $119,691(where $119,691 includes cost of footings, including labor, materials and misc costs (90%) and overhead and profit (10%)The easiest way to check NCARB calculations:$108,810 / $119,691 = 0.90909090909, or 91% and not 90%Now let's check overhead and profit for NCARB suggested final cost (10% of final cost should be overhead and profit):$119,691 x .1= $11,969The summary:In NCARB ARE handbookcase, contractor received $119,691  108,810 = $10,881 instead of $12,090 (underpaid $1,209) or only 9% profitBasically, this is an equation for profit and overhead:$108,810 / ((100% target profit%)/100) = $108,810 / .9 = $120,900And this is an equation for mark up:$108,810 x (100% + target profit%) = $108,810 x 1.1 = $119,691 
I am also confused by this question.
NCARB When we calculate for architect's profit in PcM, let's say 20%, usually we use multiply 0.2 only if the profit is already included in the given fee. We would use divide 0.8 when the fee is not yet included the profit.
So as this question, $108,810 simply is the sum of material and labor and does not account for the desired 10% profit. Like Valerie Galchenko stated, shouldn't we use divide 0.9 instead?

Prior to obtaining my M.Arch degree, I worked as a senior analyst for number of financial institutions, and my opinion is that ARE handbook solution DOES NOT show any understanding of the difference between MARK UP and a PROFIT MARGIN.
Steps to calculate a percentage based profit margin (let's say, based on 10% profit):
 1. Use 10% in its decimal form, which is 0.1.
 2. Subtract 0.1 from 1 to get 0.9
 3. Divide the original price (sum, number) by 0.9
 4. The resulting number is how much you should charge for a 10% profit margin. Mark ups are always higher than their corresponding profit margins (as we can see in NCARB case, mark up is 10% when profit is only 9.09%). I think NCARB needs to revisit the question and give the credit to all applicants who answered this question during the exam correctly. Thank you for reading.

NCARB: Which of these answers is considered correct? The $19,691 per the ARE Handbook, or the $20,900 per the method described by Valerie and the AHPP (p.415 15th E)? I understand the difference between the calculations and am disconcerted by the seemingly incorrect answer given in the ARE Handbook. If a problem like this occurs on the exam and I answer per the AHPP's method for calculating profit, will it be counted incorrect? Thank you.

Hi Bruce,
Thank you for replying. NCARB made an obvious mistake on ARE book demonstration exam and simply keeps ignoring ARE candidates questions. Please check this post as well:
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