The relationship formula is called other things, but I think if you aren’t a math person, it is better to describe what it is doing. I use it a lot: figuring new stitch counts, finding size relationship numbers, calculating yardage, weight, or other measurements. I use it so much I should have a spreadsheet for it, but I don’t. I love jotting it down on a piece of paper. For some things I can do it on my calculator without writing it down, but usually I need the visual aid. If you go back to some earlier math-learning experience you may recognize it as the cross-multiply and divide formula.

### Why is the Relationship Formula So Cool?

The relationship formula is a simple-to-learn formula that has lots of applications. In knitting, here are a few:

- When you need to change gauge,
- When you want to know the weight or yardage based on some other weight or yardage,
- When grading to other sizes,
- To calculate the time needed,
- When making other substitutions.

You can take this outside your knitting and calculate how long it takes you to walk a distance, how to enlarge or expand a recipe, and lots of other applications you will think up.

The coolness is because the relationship formula allows you to figure out how many repeats you could get if a pattern that used 75g of yarn in 18 pattern repeats, repeated 4 times, and if you wanted to use all 100g of the same yarn.

### Setting up the Relationship Formula

I used a scarf I unearthed recently and am in love with. It is knit horizontally (cast on a bunch of stitches). The stitch pattern is 10 stitches wide. The stitch pattern has 8 rows, and the rows are repeated 4 times. There are other details, but that is enough to get going.

original yarn weight new yarn weight

________________ equals ________________

original pattern repeats new pattern repeats

- 75g is the original weight
- (18 x 4) original pattern repeats
- new yarn weight is 100g
- what we don’t know is how many new repeats

The formula becomes

75g 100g

_________ equals _______

72 repeats X repeats

We are solving for X repeats.

**Cross multiply: ** 75g by X repeats you do not get a useable number, so let’s start by multiplying 72 by 100. The result is 7200.

**And Divide:** Divide 7200 by 75. The result is 96 repeats.

**Check? ** Multiply 96 by 75g and you will get 7200!

Take that a step further and divide 96 by 4 (the number of vertical pattern repeats). The answer is 24. So you could make the scarf 24 horizontal repeats to use the 100g.

### Other Uses For The Relationship Formula

What if you wanted to make this same scarf, but use a different weight yarn? Instead of 4 stitches to the inch, let’s say you got 4.5 stitches per inch in the stitch pattern with the new yarn. Use the same formula to find out how many stitches you need to cast on. Let’s say that the original cast on is 188 stitches total.

We know

- the original stitch gauge,
- the original cast on number,
- the new stitch gauge
- and we want to know the new cast on number.

4 sts 4.5 sts

_______ equals _______

CO188 COX

**Cross multiply: ** 188 and 4.5 to yield 846.

**And Divide:** 846 by 4 to yield 211.5. Obviously you will need to round to get a whole number, as well as one that fits the pattern setup. Your scarf will be approximately the same horizontal length. [212 is the next whole number. If you take 8 edge stitches away you end up with 204 stitches, which doesn’t fit the 10-stitch pattern multiple.]

**Check? ** Multiply 211.5 by 4 sts and you will get 846!

The secret lies in the information that you have and what will allow you to find the new answer. It isn’t always obvious. Let’s say you want to know the number of repeats to do to get approximately the same 5″.

What do you know?

- original row gauge (6 rows/in)
- new row gauge (7 rows/in)
- number of row repeats done (4)
- rows in pattern (8)
- total depth excluding trim (5″)
- and we want to get 5″ in the new yarn, so that isn’t what we’re solving for.

Let’s figure out using 32 (8 rows of pattern by 4 repeats) and old and new row gauge.

32 pattern rows X rows

_______ equals _______

6 rows/in 7 rows/in

**Cross multiply: ** 32 x 7 to get 224.

**And Divide:** 224 by 6 to yield 37.33. Since we have an 8 row pattern you will likely choose to do 40 rows (or 5 repeats) and your scarf will be a little deeper.

**Check? ** Multiply 37.33 by 6 rows and you will get 223.98!

You could also set this up in this manner:

6 rows 32 pattern rows

_______ equals _______

7 rows X pattern rows

The relationship formula has the same information, but it is laid out differently. Sometimes looking at the knowns in another way helps me see what I need to put in to get my unknown.

Finally, because of how the pattern is written, I’m going to assume that I need an even number of repeats horizontally to match the original number.

This is what we know:

- Original stitch count (CO188)
- Edge stitches (8 sts)
- Original repeats (18)
- Original repeats are even
- New stitch count (CO212)*

This is what we have:

18 repeats X repeats

_______ equals _______

(188-8)=180 stitches 204 stitches

**Cross multiply: ** 18 x 204 to get 3672.

**And Divide:** 3672 by 180 for 20.4 repeats. This doesn’t quite satisfy our need for an even repeat, but I’m pretty close and I can see that if I go to 20 repeats (20 x 10 = 200) I’m keeping close to the original stitch number calculation, using a multiple of 10, and preserving the odd/even relationship.

**Check? ** Multiply 180 by 20.4 repeats and you will get 3672!

See why the Relationship Formula is so cool? I’ve solved multiple things and kept an eye on the bigger picture of relationships to hone in on what is a workable result.

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