Real estate math

1.1 · Splitting a commission

Price, then brokerages, then agent

Work from the top down: the commission is the sale price times the rate; it is split between the listing and selling brokerages; each brokerage then splits its share with its agent.

agent's pay = price × rate × brokerage share × agent split

Why: The agent is paid by their own broker, never directly by the seller, so the agent's cut always comes out of the brokerage's share.

Commission: $425,000 × 6% = $25,500. Split 50/50 between the brokerages, each side gets $12,750.

The listing agent is on a 70/30 split with their broker: $12,750 × 70% = $8,925 to the agent, $3,825 to the broker.

Watch out: Apply the agent's split to the brokerage's share, not to the whole commission. 70% of the full commission would be more than the listing side ever received.

1.2 · Prorating property tax

Who owned the house on which days

Divide the annual amount by the days in the year, then multiply by the days each party owned the property. When taxes are paid in arrears, the seller has not paid their share yet, so it is a debit to the seller and a credit to the buyer.

seller's share = annual tax ÷ 365 × days the seller owned it

Why: The buyer will pay the whole year's bill later, including days the seller lived there.

The year's tax is $5,475, or $15.00 a day. Closing is June 15 and the seller is charged through the day of closing, so the seller's share runs January 1 through June 15: 166 days.

166 × $15.00 = $2,490, charged to the seller and credited to the buyer. The buyer's 199 days make up the rest.

Watch out: Read which convention the question states. If the buyer owns the day of closing instead, the seller has 165 days, $2,475. A 360-day banker's year with 30-day months moves the answer too, and February has 28 days unless the year is a leap year.

1.3 · The seller's net

Start at the price and subtract what the seller pays

The seller's net is the sale price less everything charged to the seller: the commission, the mortgage payoff, the seller's closing costs and any prorations that go against the seller.

net = price − commission − payoff − closing costs − seller debits

Why: The settlement statement lists credits and debits to the seller; the net is what is left.

$425,000 − $25,500 commission − $280,000 payoff − $4,250 closing costs − $2,490 tax proration.

The seller walks away with $112,760.

Watch out: The buyer's loan and the buyer's closing costs never come out of the seller's net. Only the seller's side of the statement counts.

1.4 · Working back to a price

What the house must sell for to net a number

Add up everything the seller must end up with or pay in fixed dollars, then divide by what is left after the commission rate.

price = (desired net + payoff + fixed costs) ÷ (1 − commission rate)

Why: The commission is a percentage of the price you are solving for, so it cannot simply be added on.

The seller wants to net $120,000, owes $280,000 and has $4,250 in fixed costs: $404,250 in all.

$404,250 ÷ 0.94 = $430,053.19. Check: 6% of that is $25,803.19, which leaves exactly $404,250.

Watch out: Multiplying by 1.06 instead gives $428,505.00, which leaves the seller short, because 6% is taken of the higher price.

1.5 · One sale, split across the street

From the closing table to each agent, and back from a net to a price

The commission is the price times the rate. The brokerages split it by their shares, and each brokerage splits its own share with its agent. The seller's net is the price less the commission, the payoff and the costs; to work back, add up what must be left and divide by what is left after the rate.

agent's pay = price × rate × brokerage share × agent split · price = (net + payoff + costs) ÷ (1 − rate)

Why: Seen as one round of stone, each slice is its share of the price: the commission is a slice that crosses the street and is cut again at each office, and working back means sizing the round so the net still fits after the commission's slice.

Work a new sale below: the commission, the two brokerages' shares, each agent's and broker's cut, the seller's net, and the price the seller would have needed for a higher net.

Watch out: The numbers below are a different sale from the one in the sections above, so work each answer rather than recalling it. This sale has no tax proration, so the net leaves that step out.