Question

Giovanni has been hired by a local clothing accessory store to make decorative belts that have green and blue beads carefully glued on them. these belts are meant for a waist size of around 28inches. each bead (green or blue) is about 1 cm wide, and Giovanni has been told to put anywhere between 70 and 74 total beats on each belt (this makes a belt with 70 to 74 cm of Beads which is 27.5 to 29.1 inches.) the store once these beads to consist of mostly green beads but also wants to have a noticeable amount of blue beads, in fact, the store has specified that the ratio of green beads to Blue beads must be at least 1: 4 and at most 1: 6

Write a system of inequalities to model this scenario.

Answer 1

The system of inequalities that model the number and types green (g)

and blue (b) beads in a belt are as follows;

• 70 < g + b < 74

• 10 < g < 14

• 56 < b < 63

• $\underline{\dfrac{1}{4} \leq \dfrac{b}{g} \leq \dfrac{1}{6}}$

How can s system of inequalities be written?

The waist size for the belt = ±28 inches

The x represent the number of beads on each belt, we have;

Number of beads per belt 70 < x < 74

Minimum ratio of blue to green beads = 1 : 4

Maximum ratio of blue to green beads = 1 : 6

Therefore;

Minimum number of blue beads = $\frac{70}{1 + 6}$ = 10

Maximum number of blue beads = $\frac{74}{1 + 4}$ ≈ 14

The number of blue beads, b, in a belt is therefore;

• 10 < g < 14

Minimum number of green beads = $\frac{4}{1 + 4}$ × 70 = 56

Maximum number of green beads = $\frac{6}{1 + 6}$ × 74 ≈ 63

The number of green beads, g, in a belt is therefore;

• 56 < b < 63

The sum of the beads on each belt = g + b = x

Therefore;

• 70 < g + b < 74

From the given maximum and minimum ratios, we have;

• $\underline{\dfrac{1}{4} \leq \dfrac{b}{g} \leq \dfrac{1}{6}}$

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