Math question about the congress. not that hard, but i dont understand it.

During a session of Congress, 4000 bills were introduced, but 2200 of them failed to pass in either house. The House of Representatives approved 1200 bills and the Senate 1500; 800 bills were signed into law by the President. How many bills were passed by both houses and vetoed by the President?

please give a thorough explanation for this.
thank you in advance! :)

draw 2 circles that intersect:

1st circle = HR
2nd circle = S

bare with me now:

let X be the no. of bills passed by the HR but NOT by the S

let Y be the no of bills passed by the S but NOT by the HR

let A be the no of bills passed by BOTH houses

i.e. in the 1st circle put an X, in the 2nd a Y and in the AREA OF INTERSECTION an A


we know that:

1. The House of Representatives approved 1200 bills

i.e. X + A = 1200

why ? because X is EXCLUSIVE to the HR, and A is both houses

2. The SENATE approved 1200 bills

i.e. Y + A = 1500

3. 4000 bills were introduced, but 2200 of them failed to pass in either house

i.e. X + Y + A = 4000 - 2200 = 1800


from the 1st 2 numbered points above : X + A + Y + A = 2700 bills

from the 3rd: X + Y + A = 1800 bills

hence: A = ? (subtract the 2 quantities above) -> A = 2700 - 1800 = 900

reminder: A is the no. of bills approved by BOTH houses (X is EXCLUSIVE to HS and Y to the SENATE)


4. 800 bills were signed into law by the President

i.e. the President approved only a certain no. of bills out of the total number that passed through BOTH houses

how many again did both houses approve ? A = 900 bills

out of these, how many were signed into the law by the President ? 800 bills

hence, how many (annoyingly repetitive right ?) were vetoed ? 900 - 800 = 100 bills

So you start off with a total of 4,000 bills presented to Congress. Of these, 2,200 of them were rejected by both houses, so that leaves us with 1,800 that were passed by at least one of the houses. The House of Rep's passed 1,200 of these, and the Senate passed 1,500. That tells us that, at most, 1,200 bills were passed by Congress, while the minimum was 900. These bills are passed on to the President, who only passed 800. That tells you that the minimum number of bills vetoed by the President was 100, and the maximum was 700. I don't believe you can get more specific than that.

Hope this helps.

it has to pass in the Senate before it has a chance to pass in the House. therefore the 1200 bills passed by the House were from the 1500 bills passed by the Senate, meaning that 1200 were passed by both the House and the Senate.
so this is indeed simple math: the 1200 passed by Congress minus the 800 signed into law leaves us with 400 that must have been vetoed by the President. the question has less to do with math than it does the mechanization of the U.S. federal gov't.