Problem
Newton's cows: grass grows while eaten
A pasture has grass standing on it and more grows each day, the same amount. It feeds 15 cows for 20 days. It also feeds 30 cows for 8 days. Find how many days it feeds 10 cows.
Reasoning · Grade 6-1 Ratio and Rate (2)
Your answer
Watch out The grass seems a fixed pile, but it keeps growing while they eat.
How to solve
Strategy Draw a Diagram — The trap in this problem is treating the pasture as a fixed pile of grass. It is not: grass is being added every day while the cows eat, so there are two hidden amounts — the grass standing at the start and the growth per day — and one scenario cannot separate them. The bar diagram is what does the separating. Drawing both scenarios as bars that start flush at the left makes the shared starting piece line up, so the two bars differ only in their right-hand growth pieces. Sliding one bar against the other and reading the leftover strip is subtraction done with your eyes: the starting grass cancels itself out and what is left is pure growth. Once x is known, the starting amount drops out of either bar, and the last part is one more subproblem.
1STEP 1
Say what one bar means
Grass eaten is the start plus the growth.
15 × 20 = (grass at the start) + x × 20
A bar split into two pieces is just a picture of 'this total is made of these two parts'. Counting the total as 15 cows x 20 days is the same repeated-addition multiplication used for any rows-and-columns count.
6.EE.B.6Draw A Diagram2STEP 2
Fill in the lower bar (part 1)
Draw the same bar for 30 cows over 8 days.
30 × 8 = (grass at the start) + x × 8
Writing the second situation in the very same form as the first is what makes the two comparable. Same shape, different numbers — nothing new has to be understood.
6.EE.B.6Draw A Diagram3STEP 3
Fill in the difference boxes (part 1)
The bars differ by 60.
x × 20 - x × 8 = x × 12 and 15 × 20 - 30 × 8 = 300 - 240 = 60
The unknown starting amount is the same in both bars, so subtracting one bar from the other erases it. That is the whole reason for lining the bars up at the left: it turns an unknown you cannot find into an unknown you do not need.
6.EE.A.3Organize Information In More Ways4STEP 4
Solve for x, the daily growth (part 2)
That is 12 days of growth, so 5 a day.
x × 12 = 60 → x = 60 ÷ 12 = 5
One quantity measured in two different ways gives an equation for free. Undoing 'times 12' with 'divided by 12' is the only step needed.
6.EE.B.7Identify SubproblemsComparing the two grazings makes the starting grass cancel and leaves the daily growth on its own.
Why?
Both scenes start with the same field, so subtracting one from the other removes the starting amount entirely.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
The grass grows by the same amount every day, so the extra days contribute that fixed step repeated, and nothing else.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
5STEP 5
Read the units of x
That growth feeds 5 cows.
x = 5 = the daily food of 5 cows
Naming the unit turns an abstract 5 into a picture. Any answer expressed in 'cow-days' can be sanity-checked against the story.
6.RP.A.2Analyze The Units6STEP 6
Find the grass standing at the start (part 3)
The grass standing at the start is 200.
(grass at the start) = 15 × 20 - 5 × 20 = 300 - 100 = 200
Once the growth piece of the bar has a number, the starting piece is just the whole bar minus that piece — plain subtraction on the picture already drawn.
6.EE.B.7Identify Subproblems7STEP 7
Check the starting amount on the other bar
The other bar also gives 200.
30 × 8 - 5 × 8 = 240 - 40 = 200 ✓
Both scenarios were used to find x, so using each of them separately to rebuild the starting grass is a genuine check, not a repeat of the same arithmetic.
6.EE.A.3Organize Information In More Ways8STEP 8
Find how long 10 cows can graze (part 4)
10 cows draw down 5 a day, so 40 days.
10 - 5 = 5 units eaten out of the store per day, 200 ÷ 5 = 40 days
Only the net loss matters. Growth is like a small refund on every day's bill, so the store runs down at the smaller rate 5, not at 10 — and 'how many days to use up 200 at 5 a day' is one division.
6.RP.A.3Identify Subproblems9STEP 9
Confirm the 40 days against the bar picture
The bar picture confirms 40 days.
10 × 40 = 400 and 200 + 5 × 40 = 200 + 200 = 400 ✓
Putting the answer back into the original bar shape is the surest check: the eaten total and the supplied total must land on the same mark.
6.EE.B.7Draw A DiagramAnswer
40 days
200 ÷ 5 = 40
Every number is in the same unit — one cow's daily meal — so all the additions and subtractions are legal. The size of the answer makes sense in two ways. First, the pattern of the given cases: 15 cows last 20 days and 30 cows last only 8, so fewer cows must last longer than 20 days, and 10 cows lasting 40 days fits. Second, the growth is worth 5 cows a day, so 10 cows are really only 5 cows' worth of drain on the standing 200 units; halving the drain roughly doubles the time compared with the 15-cow case, and 20 days doubling to 40 is exactly what happened. It is also worth noticing what the wrong model predicts: if the pasture were a fixed pile, 15 cows for 20 days would be 300 units and 10 cows would last 30 days — but that same fixed pile would have to be 240 units to match the 30-cow case, a contradiction. The two given scenarios simply cannot both be true unless grass is growing. Finally, the answer is a whole number of days and the balance check 10 x 40 = 200 + 5 x 40 comes out exact, so the grass runs out on the day, not part-way through one.
Takeaway
The grass keeps growing while the cows eat, so line up the two stories as bars and subtract — the part you cannot find cancels itself out.
- Say what one bar means
- Fill in the lower bar (part 1)
- Fill in the difference boxes (part 1)
- Solve for x, the daily growth (part 2)
- Read the units of x
- Find the grass standing at the start (part 3)
- Check the starting amount on the other bar
- Find how long 10 cows can graze (part 4)
- Confirm the 40 days against the bar picture