Mathematic Problemsums ?
10 stars for Best Solutions and Explanation
Q1. Ivy bought some marbles and gave half of them to Joan. Joan bought some stamps and have half of them to Ivy. Ivy gave away 8 stamps and Joan gave away 12 marbles. The ratio of the number of stamps to the number of marbles Ivy had left then became 1:5 and the ratio of the number of stamps to the number of marbles Joan had left became 1:3.
How many stamps did Joan buy?
Q2.Mrs Ang and Mrs Bala were supposed to bake 280 cupcakes for a party altogether.In the end, Mrs Ang and Mrs Bala baked 328 cupcakes instead. If Mrs Ang baked 1/5 more cupcakes than what she should and Mrs Bala baked 3/20 more cupcakes than she should,how many cupcakes were each of them suppose to bake at first?
10 stars for Best Solutions and Explanation
Q1. Ivy bought some marbles and gave half of them to Joan. Joan bought some stamps and have half of them to Ivy. Ivy gave away 8 stamps and Joan gave away 12 marbles. The ratio of the number of stamps to the number of marbles Ivy had left then became 1:5 and the ratio of the number of stamps to the number of marbles Joan had left became 1:3.
How many stamps did Joan buy?
Q2.Mrs Ang and Mrs Bala were supposed to bake 280 cupcakes for a party altogether.In the end, Mrs Ang and Mrs Bala baked 328 cupcakes instead. If Mrs Ang baked 1/5 more cupcakes than what she should and Mrs Bala baked 3/20 more cupcakes than she should,how many cupcakes were each of them suppose to bake at first?
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Q1. Let "m" represent the number of marbles Ivy bought. Let "s" represent the number of stamps Joan bought.
.. stamps given to Ivy = s/2
.. stamps Joan has left = s - s/2 = s/2
.. stamps Ivy has left after giving away 8 = s/2 - 8
.. marbles given to Joan = m/2
.. marbles Ivy has left = m - m/2 = m/2
.. Ivy's stamps to marbles = (s/2 - 8)/(m/2) = 1/5
.. marbles Joan has left after giving away 12 = m/2 - 12
.. Joan's stamps to marbles = (s/2)/(m/2 - 12) = 1/3
Our two equations in two unknowns are
.. (s/2 - 8)/(m/2) = 1/5 … the first equation
.. (s/2)/(m/2 - 12) = 1/3 … the second equation
multiplying the first by 5(m/2), we have
.. 5(s/2 - 8) = m/2
.. 5(s/2) - 40 = m/2 … eliminate parentheses
.. 5s - 80 = m … multiply by 2. This expression for m is used below.
Multiplying the second by 3(m/2 - 12), we have
.. 3(s/2) = m/2 - 12
.. 3s = m - 24 … multiply by 2
.. 3s = (5s - 80) - 24 … substitute for m from the result above.
.. 0 = 2s - 104 … subtract 3s, collect terms
.. 104 = 2s … add 104
.. 52 = s … divide by 2
Joan bought 52 stamps
Q2. Let "a" be the number of cupcakes Mrs Ang was supposed to bake, and "b" be the number of cupcakes Mrs Bala was supposed to bake. Then we have
.. a + b = 280
.. a(1 + 1/5) + b(1 + 3/20) = 328
Multiplying the second equation by 20, we have
.. 24a + 23b = 20*328
Subtracting 23 times the first equation, we have
.. (24a + 23b) - (23a + 23b) = (20*328) - (23*280)
.. a = 120 … collect terms, evaluate
.. b = 280 - a = 160
Mrs Ang was supposed to bake 120 cupcakes; Mrs Bala was supposed to bake 160 cupcakes.
.. stamps given to Ivy = s/2
.. stamps Joan has left = s - s/2 = s/2
.. stamps Ivy has left after giving away 8 = s/2 - 8
.. marbles given to Joan = m/2
.. marbles Ivy has left = m - m/2 = m/2
.. Ivy's stamps to marbles = (s/2 - 8)/(m/2) = 1/5
.. marbles Joan has left after giving away 12 = m/2 - 12
.. Joan's stamps to marbles = (s/2)/(m/2 - 12) = 1/3
Our two equations in two unknowns are
.. (s/2 - 8)/(m/2) = 1/5 … the first equation
.. (s/2)/(m/2 - 12) = 1/3 … the second equation
multiplying the first by 5(m/2), we have
.. 5(s/2 - 8) = m/2
.. 5(s/2) - 40 = m/2 … eliminate parentheses
.. 5s - 80 = m … multiply by 2. This expression for m is used below.
Multiplying the second by 3(m/2 - 12), we have
.. 3(s/2) = m/2 - 12
.. 3s = m - 24 … multiply by 2
.. 3s = (5s - 80) - 24 … substitute for m from the result above.
.. 0 = 2s - 104 … subtract 3s, collect terms
.. 104 = 2s … add 104
.. 52 = s … divide by 2
Joan bought 52 stamps
Q2. Let "a" be the number of cupcakes Mrs Ang was supposed to bake, and "b" be the number of cupcakes Mrs Bala was supposed to bake. Then we have
.. a + b = 280
.. a(1 + 1/5) + b(1 + 3/20) = 328
Multiplying the second equation by 20, we have
.. 24a + 23b = 20*328
Subtracting 23 times the first equation, we have
.. (24a + 23b) - (23a + 23b) = (20*328) - (23*280)
.. a = 120 … collect terms, evaluate
.. b = 280 - a = 160
Mrs Ang was supposed to bake 120 cupcakes; Mrs Bala was supposed to bake 160 cupcakes.