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William has a 26 litre glass tank first he wants to put some marbles in it all of the same volume then he wants to fill the tank with water until it's completely full if he uses 85 marbles he will have to add 20.9 liters of water what is the volume of each marble all right so let's think about it the volume of the marbles plus the volume of the water are going to be equal to the volume of the tank they're going to fill up the tank let me write that down so the volume let me write V V and I'll write a little subscript m here this is the volume of the marbles plus the volume of the water plus the volume of the water are going to be equal to the volume of the tank are going to be equal to the volume of the tank now we don't know what the what the volume of 85 marbles is but we know this right over here this is going to be 85 times the volume of one marble so this is volume of all the marbles and maybe let me I'll just write it out volume volume of one marble one marble I'll write it out I could have put a variable in there but just to make it clear so 85 times the volume of one marble that's going to be the total volume of the marbles plus the volume of water they tell us what the volume of water is twenty point nine liters of water plus twenty point nine is going to be equal to the volume of the entire tank it's a 26 liter glass tank so it's going to be equal to 26 remember the whole thing that's going on here is he wants to fill up the whole tank he puts some some marbles in that have some volume and then whatever is left he fills it with water so these two volumes combined have to add up to the volume of the entire tank now what is though now we could just solve for the volume of one marble and actually maybe I'll I'll just let's just call this the variable M let's just call this this let's call it M for the volume of one marble so we get so we get 85 we get 85 times m-maybe I'll just do this in one color for the sake of time plus twenty point nine is equal to 26 now to solve for M we could subtract 20 point 9 from both sides so subtract 20 point 9 subtract 20 point 9 from both sides and we get 85 times the volume of one marble is equal to let's see 26 minus 21 would have been 5 so this is going to be 5 point no this is going to be right it says this is going to be 5 point 1 it's going to be 5 five point one or we can say the volume of each marble if we divide both sides by 85 is five point one liters over 85 and I could I could calculate let's see let's well let me let me just get a calculator out for this we're allowed to use calculators on this one and so we get five point one divided by 85 is equal to 0.06 liters is equal to 0.06 so what is the volume of each marble we just figured out 0.06 liters and that how much water is necessary if William uses 200 200 marbles so let's think about it how much water would be necessary let's see the volume let's just say W is the amount of water that's necessary and it's going to be a function of the number of marbles the number of marbles and let's see if if you have if you have no marbles you're going to have to put in 26 liters you have to fill up the whole glass tank the whole volume so it's going to be equal to 26 minus the volume of the marbles well what's the volume of the marbles going to be well it's going to be the volume of each marble it's going to be 0.06 times the number of marbles you have times the number of marbles you have just you just like that we've described a linear function that tells us how much water we have to fill as a function of marbles it's going to be the total volume of the tank minus the volume of the marbles and the volume of the marbles is the volume per marble times the number of marbles so if they're saying how much water we need for 200 marbles that's going to be w double the amount of water we have to fill as a function of marbles we're now the number of marbles is 200 that's going to be equal to 26 minus 0.06 times 200 times 200 and what's that going to be well let's see 6 times this is 6 this is 6 hundredths 6 hundredths times a hundred would be 6 but now we're multiplied by 200 so there's going to be 12 so this thing right over here this thing right over here is going to be equal to 12 and then 26 minus 12 is 14 so all of this let me divided like this all of this is going to be equal to 14 so how much water is necessary if William uses 200 marbles 14 liters of water 26 minus 0.06 times 200