Nicole makes a keychain by using yellow, red and green beads. The number of red beads used is more than twice the number of yellow beads by 10, while the number of green beads used is more than one-third the number of yellow beads by 7. It is known that each yellow bead, red bead and green bead weigh 2 g, 1 g and 6 g respectively. The total weight of the beads used is 160 g.
a) Find the number of yellow beads used.
b) Nicole claims that she makes the keychain with not more than 80 beads. Do you agree? Explain your answer.
A phrase such as
"The number of red beads used is more than twice the number of yellow beads by 10"
seems to cause a lot of difficulty, since students often get confused where to add or subtract the difference given.
Perhaps this will help:
state the given as an inequality,
r > 2y by 10
now "balance" that inequation, that is, add 10 to the lesser side:
r = 2y + 10
similarly :
"the number of green beads used is more than one-third the number of yellow beads by 7"
g > (1/3)y by 7 ----> g = (1/3)y + 7
3g = y + 21
so 2y + r + 6g = 160
2y + (2y+10) + 2(3g) = 160
2y + 2y + 10 + 2y + 42 = 160
6y = 108
y = 18
so she uses 18 yellows
r = 2(18) + 10 = 46
3g = 18+21 = 39
g=13
she uses 18 yellows, 46 reds, and 13 greens
What is your conclusion for b) ?
Check my answer to make sure all conditions are met.
remember we are talking about the number of beads and the weight of the beads.
School
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume that the number of yellow beads used is Y, the number of red beads used is R, and the number of green beads used is G.
According to the problem, the number of red beads used is more than twice the number of yellow beads by 10. This can be written as:
R = 2Y + 10 (Equation 1)
Also, the number of green beads used is more than one-third the number of yellow beads by 7. This can be written as:
G = (1/3)Y + 7 (Equation 2)
The weight of each yellow bead is 2g, each red bead is 1g, and each green bead is 6g. We know that the total weight of the beads used is 160g. So we can write the weight equation as:
2Y + 1R + 6G = 160 (Equation 3)
Now we have a system of three equations with three unknowns. We can solve this system to find the values of Y, R, and G.
To find the number of yellow beads used (Y), we can substitute equations 1 and 2 into equation 3.
Substitute R from equation 1 into equation 3:
2Y + 1(2Y + 10) + 6G = 160
Simplify and combine like terms:
2Y + 4Y + 10 + 6G = 160
Combine Y terms:
6Y + 10 + 6G = 160
Subtract 10 from both sides:
6Y + 6G = 150 (Equation 4)
Substitute G from equation 2 into equation 4:
6Y + 6((1/3)Y + 7) = 150
Simplify and combine like terms:
6Y + 2Y + 42 = 150
Combine Y terms:
8Y + 42 = 150
Subtract 42 from both sides:
8Y = 108
Divide both sides by 8:
Y = 13.5
Since we cannot have a fraction of a bead, we can conclude that Nicole used 13 yellow beads (Y = 13).
Now let's find out if Nicole's claim of not using more than 80 beads is correct.
To calculate the total number of beads used, we can substitute Y = 13 into equations 1 and 2:
R = 2Y + 10
R = 2(13) + 10
R = 26 + 10
R = 36
G = (1/3)Y + 7
G = (1/3)(13) + 7
G = 4.33 + 7
G ≈ 11.33
The total number of beads used is:
Total beads = Y + R + G
Total beads = 13 + 36 + 11.33
Total beads ≈ 60.33
So Nicole used approximately 60.33 beads, which is less than 80 beads. Therefore, her claim is correct.