7/8 students own a bike . 72 is the difference between the ones that own a bike and ones that do not. how many were surveyed?

X + y is the number of students who were surveyed. x is the number who own a bike. y is the number who don't. the ratio of the ones who own a bike and the ones who were surveyed is x / (x + y) that ratio is 7/8. you get x / (x + y) = 7/8 multiply both sides of this equation by (x + y) to get x = (x + y) * 7/8 simplify to get x = 7/8 * x + 7/8 * y subtract 7/8 * x from both sides of this equation to get: 1/8 * x = 7/8 * y solve for x to get x = 7 * y this means that the number of students who own a bike is 7 times the number of students who don't. divide both sides of that equation by y to get x/y = 7/1. that means that the ratio of students who own a bike to the number of students who don't is equal to 7/1. the problem states that the difference between the number who own a bike and the number who don't is 72. this means that x - y = 72 from the equation of x = 7 * y, replace x with 7 * y in the equation of x - y = 72 to get: 7 * y - y = 72 simplify to get 6 * y = 72 divide both sides of that equation by 6 to get y = 72/6 = 12 this means that the number of students who do not own a bike is 12. since x = 7 * y, then x = 7 * 12 = 84. this means that the number of students who own a bike is 84. since the number of students is x + y, this means that the number of students who were surveyed is 84 + 12 = 96. let's see if this is right. 96 students were surveyed. 84 own a bike and 12 don't. the difference between the ones who own a bike and the ones who don't is 84 - 12 = 72. the ratio of the ones who own a bike and the number surveyed is 84/96 = 7/8. solution is 96 students were surveyed.