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Secondary 1 | Maths
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For Qn 3
R/B = 5/2
R is the number of red ants.
B is the number of black ants.
R - .3R is equal to .7R.
let x equal the ratio increase in the number of black ants so that there is an equal number of red ants and black ants in the colony after the number of red ants has been reduced by 30%.
the new ratio is .7R / (B + xB) = 1
since R = 5B/2, the equation becomes:
(.7 * 5B/2) / (B + xB) = 1
simplify to get:
(3.5B/2) / (B*(1+x)) = 1
simplify further to get:
(3.5/2) / (1 + x) = 1
multiply both sides of the equation by (1 + x) to get:
3.5/2 = 1 + x
simplify to get:
1.75 = 1 + x
solve for x to get:
x = 1.75 - 1 = .75
the formula of .7R / (B + xB) = 1 becomes:
.7R / (1.75B = 1.
if you increase the value of B by a factor of .75, that's a 75% increase.
your original ratio is R/B = 5/2.
your new ratio is .75R / 1.75B = 1/1.
assume the total number of ants in the colony is 420.
you get R + B = 420
solve for R to get:
R = 420 - B.
the original ratio is R/B = 5/2
when R = 420 - B, the ratio becomes:
(420 - B) / B = 5/2.
that becomes:
420/B - B/B = 5/2, which becomes:
420/B - 1 = 5/2.
add 1 to both sides of the equation to get:
420/B = 1 + 5/2
simplify to get:
420/B = 7/2
solve for B to get:
B = 420 / (7/2) = 120.
since R + B = 420, then R must be equal to 300.
the original ratio becomes:
R/B = 300 /120 = 5/2.
the original number of red ants is 300 if the total number of ants is 420.
when you reduce the number of red ants by 30%, you get:
.7 * 300 = 210.
that's the number of red ants after they were decreased by 30%.
you start with 300 red ants and 120 black ants.
70% red ants is equal to 210.
1.75 * black ants is equal to 1.75 * 120 = 210.
the ratio of .75R / 1.75B becomes 210/210 = 1/1.
it all checks out.
your solution is:
(a) If the number of red ants decreases by 30%, find the percentage increase in the number of black ants so that there is an equal number of red ants and black ants in the colony.
increasing the number of black ants by 75% gets you a ratio of .75R / 1.75B = 1/1.
(b) If the original number of ants in the colony is 420, find the number of red ants after the decrease.
the number of red ants in the colony after the 30% decrease is 300 * .75 = 210.
R/B = 5/2
R is the number of red ants.
B is the number of black ants.
R - .3R is equal to .7R.
let x equal the ratio increase in the number of black ants so that there is an equal number of red ants and black ants in the colony after the number of red ants has been reduced by 30%.
the new ratio is .7R / (B + xB) = 1
since R = 5B/2, the equation becomes:
(.7 * 5B/2) / (B + xB) = 1
simplify to get:
(3.5B/2) / (B*(1+x)) = 1
simplify further to get:
(3.5/2) / (1 + x) = 1
multiply both sides of the equation by (1 + x) to get:
3.5/2 = 1 + x
simplify to get:
1.75 = 1 + x
solve for x to get:
x = 1.75 - 1 = .75
the formula of .7R / (B + xB) = 1 becomes:
.7R / (1.75B = 1.
if you increase the value of B by a factor of .75, that's a 75% increase.
your original ratio is R/B = 5/2.
your new ratio is .75R / 1.75B = 1/1.
assume the total number of ants in the colony is 420.
you get R + B = 420
solve for R to get:
R = 420 - B.
the original ratio is R/B = 5/2
when R = 420 - B, the ratio becomes:
(420 - B) / B = 5/2.
that becomes:
420/B - B/B = 5/2, which becomes:
420/B - 1 = 5/2.
add 1 to both sides of the equation to get:
420/B = 1 + 5/2
simplify to get:
420/B = 7/2
solve for B to get:
B = 420 / (7/2) = 120.
since R + B = 420, then R must be equal to 300.
the original ratio becomes:
R/B = 300 /120 = 5/2.
the original number of red ants is 300 if the total number of ants is 420.
when you reduce the number of red ants by 30%, you get:
.7 * 300 = 210.
that's the number of red ants after they were decreased by 30%.
you start with 300 red ants and 120 black ants.
70% red ants is equal to 210.
1.75 * black ants is equal to 1.75 * 120 = 210.
the ratio of .75R / 1.75B becomes 210/210 = 1/1.
it all checks out.
your solution is:
(a) If the number of red ants decreases by 30%, find the percentage increase in the number of black ants so that there is an equal number of red ants and black ants in the colony.
increasing the number of black ants by 75% gets you a ratio of .75R / 1.75B = 1/1.
(b) If the original number of ants in the colony is 420, find the number of red ants after the decrease.
the number of red ants in the colony after the 30% decrease is 300 * .75 = 210.
thanks !
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