# Loan math problem

There was a 9gag post some time ago with the following problem:

A car dealer offers Lachlan a car that is priced at \$21800, but he only has a \$600 deposit. Lachlan's trade-in is actually worth \$500. The finance company insists on a 10% deposit for a 22% flat rate deal, so the dealer inflates the prices of both the trade-in and the car purchased. What are the inflated prices and what is the monthly repayment to pay off the car in 4 years?

I don't understand the price inflation part. Let `D = \$600`, `P = \$21800` and `T = \$500`. Does it mean then that `D + T*R = 10%*R*P` where `R` is the inflation rate? In that case `R = 0.357` and we have deflation instead of inflation. I have a feeling I am missing something here. Can someone explain?

The price inflation isn't a percentage, it's a fixed amount. If the dealer adds \$R to the price of both the trade-in and the purchased car, then everyone ends up with the right amount of money in their pockets.

So your formula should be: `D + T + R = 0.1 * (P + R)`

• If you inflated all the numbers by a percentage -- (D+T)*R=0.1*(P+R) -- then the problem has no solution, no matter what you multiply them all by, the ratio is going to be the same! And even if the loan company accepted it for whatever reason, the customer would be insane to accept it: e.g. doubling the price and doubling the deposit + trade-in would double the amount you have to finance, hardly a good deal. Okay, I guess you can't multiply the down payment. But in this case, using your formula, there's no way the dealer is going to cut the price by almost 2/3 just to meet the loan criteria.
– Jay
Commented Jul 25, 2014 at 13:51

Lachlan has \$600 cash and a car worth \$500. That's \$1,100. The new car is priced at \$21,800. Lachlan needs a loan for \$20,700. However, the finance company insists that the buyer must pay a 10% deposit, which is \$2,180. Lachlan only has \$1,100, so no loan.

The car dealer wants to make a sale, so suggests some tricks. The car dealer could buy Lachlan's old banger for \$1,500 instead of \$500, and sell the new car for \$22,800 instead of \$21,800. Doesn't make a difference to the dealer, he gets the same amount of cash. Now Lachlan has \$600 cash and \$1,500 for his car or \$2,100 in total. He needs 10% of \$22,800 as deposit which is \$2,280. That's not quite there but you see how the principle works. Lachlan is about \$200 short. So the dealer adds \$1,200 to both car prices. Lachlan has \$600 cash and a car "worth" \$1,700, total \$2,300. The new car is sold for \$23,000 requiring a \$2,300 deposit which works out exactly.

How could we have found the right amount without guessing?

Lachlan had \$1,100. The new car costs \$21,800. The dealer increases both prices by x dollars. Lachlan has now \$1,100 + x deposit. The car now costs \$21,800 + x. The deposit should be 10%, so \$1,100 + x = 10% of (\$21,800 + x) = \$2,180 + 0.1 x.

\$1,100 + x = \$2,180 + 0.1 x : Subtract \$1,100

x = \$1,080 + 0.1 x : Subtract 0.1 x

0.9 x = \$1,080 : Divide by 0.9

x = \$1,080 / 0.9 = \$1,200

The dealer inflates the cost of the new car and the value of the old car by \$1,200. Now that's the theory. In practice I don't know how the finance company feels about this, and if they would be happy if they found out.