Cameron, hannah and daniel are swimming laps to raise money for charity together the three swimmers swam a total of 1875 laps hannah swam twice as many laps as daniel cameron swam 125 more laps than hannah how many laps did each person swim

Hannah swam 700 laps, Cameron swan 825 laps, and Daniel swam 350 laps

Step-by-step explanation:

Step 1:  First we must represent this situation as a mathematical expression.  We will assign variables for the three swimmers.  C for Cameron, H for Hannah, and D for Daniel.  

All three swam a total of 1875 laps.  That is the sum of their efforts, we can write this as

      C + H + D = 1875

Step 2:   We want to use the rest of the information to create mathematical equations.  

"hannah swam twice as many laps as daniel"  can be written as

 H = 2D     (Hannah's effors is twice Daniel's effort)

"cameron swam 125 more laps than hannah" can be written as

C = H + 125   (Cameron's effort was 125 more laps than Hannah's effort)

Step 3:  We can see that both equations have H in them.  So we want to   rewrite them both so they are in terms of H, that means solving the other variable.

  H = 2D  becomes   H/2 = D  (this is now solved in terms of D)

and  

C = H + 125  is already solved in terms of C

Step 4:  Substitute the new equations in to the original equation, now there will only be one variable.  

  C + H + D = 1875  becomes   H + 125 + H + H/2 = 1875

Step 5:  Now combine like terms and solve for H

2H + H/2 = 1750   (subtract 125 from both sides, combine H terms)

5H/2 = 1750  (combine 2H and H/2)

        2H  +  H/2   becomes   4H/2  +  H/2  which simplifies to  5H/2

Now solve the equation for H

5H = 3500  (multiply both sides by 2 to get rid of the fraction)

H = 750  (divide both sides by 5 is get H by itself)

So Hannah swan 750 laps.  

Step 5:  We now need to plug 750 in the both equations from step 3 to determine how many laps Cameron and Daniel swam

    For Daniel

     H/2 = D  becomes  700/2  = D   which simplifies to 350 = D

    So Daniel swam 350 laps

    For Cameron

    C = H + 125 becomes C = 700 + 125   which simplifies to C = 825

   So Cameron swam 825 laps

Step 6:  Checking our work

Our equation are:

H = 700, C = 825, and D = 350

700 + 825 + 350 does equal 1875, so we have the correct values.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.