Video: AQA GCSE Mathematics Higher Tier Pack 4 β€’ Paper 1 β€’ Question 18

In the Venn diagram, πœ‰ = 60 musicians. 𝐢 = musicians that play classical music, 𝐽 = musicians that play jazz music. 42 musicians play only classical or only jazz music. 4/7 of these 42 musicians play only classical music. The number of musicians that play jazz music is three times the number of musicians who play neither classical nor jazz music. Complete the Venn diagram.

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Video Transcript

In the Venn diagram, πœ‰ equals 60 musicians. 𝐢 equals musicians that play classical music. 𝐽 equals musicians that play jazz music. 42 musicians play only classical or only jazz music. Four-sevenths of these 42 musicians play only classical music. The number of musicians that play jazz music is three times the number of musicians who play neither classical nor jazz music. Complete the Venn diagram.

We know that 42 musicians play only classical or only jazz music. On the Venn diagram, the space shaded yellow would represent the musicians that only play classical music. And the space shaded in pink represents the musicians that only play jazz music. The middle section is the space for musicians that play both classical and jazz music. We’ll consider this space a little bit later. But for now, we know that this lowercase 𝑐 plus lowercase 𝑗 is equal to 42.

Our next statement says four-sevenths of these 42 musicians play only classical music. We find the musicians who play only classical music by multiplying four-sevenths and 42. We can reduce four-sevenths times 42 by cross-canceling. Seven goes into seven one time and then to 42 six times. Four times six will equal the only classical musicians, which equals 24. We can write 24 in the space for the musicians that only play classical music.

Going back to the first statement, if we know that 42 musicians play only classical or only jazz music, we can take those 42 musicians. Subtract the 24 that play only classical music. And that will tell us how many musicians play only jazz music. 42 minus 24 equals 18. 18 musicians played only jazz music.

The next bit of information says the number of musicians that play jazz music is three times the number of musicians who play neither classical nor jazz music. At this point, our Venn diagram is still missing two pieces of information. We can let π‘₯ represent the number of musicians that play classical and jazz music. And we’ll let 𝑦 equal the musicians that play neither classical nor jazz music.

The number of musicians that play jazz music is equal to π‘₯, the number of students [musicians] that played both classical and jazz, plus 18, the students [musicians] who only played jazz. We can represent the total number of students [musicians] who played jazz with the expression π‘₯ plus 18. And we know that this is equal to three times the number of musicians who played neither classical nor jazz, three times 𝑦.

We can rewrite that a little bit more simply as three 𝑦. We’ll call this equation one. π‘₯ plus 18 equals three 𝑦. But in order to solve for our π‘₯- and 𝑦-variables, we’ll need another equation. If we consider the total number of musicians equals 60, we can say that 60 equals the 42 musicians that play only classical or only jazz plus π‘₯, the musicians that play both classical and jazz, plus 𝑦, the musicians that play neither classical nor jazz.

We can subtract 42 from both sides of the equation. 60 minus 42 equals 18. 42 minus 42 cancels out. 18 equals π‘₯ plus 𝑦. At this point, we’ll solve for either our π‘₯- or our 𝑦-value. I’m gonna choose to solve for 𝑦. We do that by subtracting an π‘₯ from both sides of the equation. 18 minus π‘₯ is 18 minus π‘₯. On the right side, π‘₯ minus π‘₯ cancels out. And we’re left with 𝑦. 𝑦 equals 18 minus π‘₯. And this is our second equation.

We’ll take our second equation and plug it in to our first. In place of 𝑦, we’ll substitute 18 minus π‘₯. π‘₯ plus 18 equals three times 18 minus π‘₯. Expanding the brackets, three times 18, three times negative π‘₯ equals negative three π‘₯. Bring down the left-hand side, π‘₯ plus 18.

In order to solve for π‘₯, we can add three π‘₯ to both sides of the equation. π‘₯ plus three π‘₯ equals four π‘₯. Bring down the plus 18. On the right side, negative three π‘₯ plus three π‘₯ cancels out, leaving us with only 54. From there, we subtract 18 from both sides of the equation, which gives us four π‘₯ equals 36. We’re nearly there.

Finally, we divide both sides by four. Four π‘₯ divided by four equals π‘₯. 36 divided by four equals nine. π‘₯ equals nine. The number of musicians who play classical and jazz music is nine. We can take what we found π‘₯ to be and plug it into the equation 𝑦 equals 18 minus π‘₯. 18 minus nine equals nine. And that means 𝑦 is also equal to nine. The number of musicians that played neither classical nor jazz music is nine.

To do a quick check, we add all four values inside the diagram: 24 plus nine plus 18 plus nine, which does equal 60.

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