Field Notesgrade 7 math
Back to Proportionality

Solution key: The number that never changes

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Camp · The number that never changes

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1 of 10 · constant-k-c01

y = 7x. What is k?

  1. In a proportional equation y = kx, k is the multiplier of x.
  2. Compare y = 7x with y = kx: the multiplier is 7.
  1. Each 1 of x gives 7 of y.
  2. So k = 7. Check: when x = 1, y = 7 × 1 = 7.

Answer: 7

2 of 10 · constant-k-c02

When x = 4, y = 20. If this is proportional, what is k?

  1. k is the output divided by the input: k = y divided by x.
  2. Here y is 20 and x is 4.
  3. 20 divided by 4 is 5, so each 1 of x matches 5 of y.
  1. Divide y by x, not x by y, or you would get the flip 4/20.
  2. So k is 5.

Answer: 5

3 of 10 · constant-k-c03

Assume y is proportional to x. When x = 8, y = 2. What is k?

  1. k = y divided by x, the amount of y for one x.
  2. y is 2 and x is 8, so split 2 into 8 equal pieces.
  3. 2 divided by 8 is 0.25, the same as 1/4.
  1. k is less than 1 because y is smaller than x.
  2. So k is 0.25.

Answer: 0.25

4 of 10 · constant-k-c04

Assume y is proportional to x. A table: (3, 12), (5, 20), (6, 24). What is k?

Table. Columns: x, y. Rows: 3, 12; 5, 20; 6, 24.
xy
312
520
624
  1. k should be the same for every pair, so divide each y by its x.
  2. 12 divided by 3 is 4.
  3. 20 divided by 5 is 4, and 24 divided by 6 is 4.
  1. All three pairs give 4, so the rate really stays the same.
  2. So k is 4.

Answer: 4

5 of 10 · constant-k-c05

A line through (0,0) and (9, 36). What is k?

  1. The given straight line goes through (0, 0), so it is proportional and has equation y = kx.
  2. Use the other point: when x is 9, y is 36.
  3. k is 36 divided by 9.
  1. 36 divided by 9 is 4, which means 4 of y for each 1 of x.
  2. So k is 4.

Answer: 4

6 of 10 · constant-k-c06

In y = kx, k = 6 and x = 11. What is y?

  1. The equation is y = kx, so multiply k by x to get y.
  2. k is 6 and x is 11.
  3. 6 times 11 is 66.
  1. You multiply because you already know the rate and the input.
  2. So y is 66.

Answer: 66

7 of 10 · constant-k-c07

In y = kx, k = 2.5 and y = 15. What is x?

  1. We know k = 2.5 and y = 15. Find x in y = kx.
  2. y is the rate times the input, so 2.5 × x = 15. Divide 15 by 2.5 to undo the multiplication.
  3. Multiply both numbers by 10 so the divisor is whole: 150 ÷ 25. 25 × 4 = 100, and 150 − 100 = 50 left.
  1. 25 × 2 = 50, so the rest is 2. 4 + 2 = 6, and x = 6.
  2. Check: 2.5 × 6 = 15, the given y.

Answer: 6

8 of 10 · constant-k-c08

Is a line through (0, 4) and (2, 10) proportional? Type yes or no.

  1. A proportional relationship has a straight-line graph through (0, 0).
  2. This line goes through (0, 4), so when x is 0, y is already 4.
  3. That 4 is a start, like a fee before anything else happens.
  1. A y-value of 4 when x = 0 means this line does not pass through (0, 0), so the relationship is not proportional.
  2. So the answer is no.

Answer: no

9 of 10 · constant-k-c09

A recipe uses 2 cups of water for every 3 cups of mix. Water is y, mix is x. What is k? Give an exact fraction.

  1. Water is y and mix is x, so k is water divided by mix.
  2. The recipe uses 2 cups of water for 3 cups of mix.
  1. 2 ÷ 3 = 2/3 cup water per cup mix. Keep the exact fraction instead of rounding its repeating decimal.
  2. So k = 2/3. Check: (2/3) × 3 cups mix = 2 cups water.

Answer: 2/3

10 of 10 · constant-k-c10

y = (3/4)x. What is k as a fraction?

  1. The equation is already y = kx.
  2. The number in front of x is the fraction 3/4.
  3. That means y is three-fourths of x, and 3/4 is the constant k.
  1. Do not flip it to 4/3, because k is the multiplier of x.
  2. So k is 3/4.

Answer: 3/4