Lesson 26 · The Stable Story · Chapter 1

The Ledger with a Missing Page

Use constant rates to decide which stable records Mia can trust.

26 · Audit27 · Build28 · Investigate29 · Present30 · Organise31 · Open

Previously, at Herr Berger's stable… Mia helped care for Blitz and Luna and worked out that the horse-care project could earn money. Herr Berger has now offered an unused field for a small rescue stable—but the old record book has gaps and conflicting numbers. Before anyone spends the budget, Mia must find which tables show a dependable constant rate.

Mia checks the old stable ledger while Blitz and Luna watch from the stall door.

A constant rate means that whenever the input changes by a certain amount, the output changes by a predictable matching amount. In this chapter, that is evidence: a table with a constant rate can support a delivery plan, a budget, or a safety decision; a table without one needs investigation.

$x: 0\to2$
$\Delta x=2$

→

$y: 5\to11$
$\Delta y=6$

→

$\frac{\Delta y}{\Delta x}=\frac62=3$

Important: compare changes, not the raw table values. If the $x$-steps are different sizes, divide each $y$-change by its matching $x$-change.

Worked example

hours $x$	0	2	4
distance $y$ (km)	1	7	13

Each $\Delta x=2$ hours and each $\Delta y=6$ km, so the rate is $6\div2=3$ km/hour. The starting value is $1$ km, so $y=3x+1$.

Read the changes

Problem 1 · The hay-delivery record
The old book claims the rescue horses received the following cumulative numbers of hay bales.
day $x$ 0 2 4 6
total bales $y$ 5 11 17 23
1. Calculate each $\Delta x$ in days.
2. Calculate each $\Delta y$ in bales.
3. Calculate the delivery rate $\frac{\Delta y}{\Delta x}$ with units.
4. Choose CONSTANT or NOT CONSTANT and state whether Mia can use this rate to plan the next delivery.
Why it matters: a dependable rate tells Mia how quickly the stored hay will grow; an inconsistent record cannot safely support a feed plan.
x-changes → y-changes → divide → decide →
Problem 2 · Battery decrease
hours $x$ 0 1 2 3
battery $y$ (%) 90 82 74 66
1. Choose INCREASES or DECREASES.
2. Calculate the rate with units.
3. Interpret the rate using “percentage points per hour”.
direction → changes → rate → sentence →
Problem 3 · The suspicious water record
The book says the trough contained the following amounts after the pump started.
minute $x$ 0 1 2 3
water $y$ (L) 2 5 9 14
1. List the three changes in water.
2. Choose CONSTANT or NOT CONSTANT.
3. Explain whether Mia should trust one rate to predict when the trough will be full.
Meaning: different interval rates suggest that the pump, measurement, or record needs checking.
differences → compare → explain →
Problem 4 · Fill a missing table value
$x$ 1 3 5 7
$y$ 8 ? 20 26
1. Calculate the constant rate using two complete columns.
2. Calculate the missing value.
3. Check the rate on both sides of the missing value.
known pair → rate → missing value → check →
Problem 5 · Table to equation
$x$ 0 2 4
$y$ 40 54 68
1. Calculate the constant rate $m$.
2. State the starting value $b$.
3. Write the equation $y=mx+b$.
rate → x=0 value → equation →

day $x$	0	2	4	6
total bales $y$	5	11	17	23

hours $x$	0	1	2	3
battery $y$ (%)	90	82	74	66

minute $x$	0	1	2	3
water $y$ (L)	2	5	9	14

$x$	1	3	5	7
$y$	8	?	20	26

$x$	0	2	4
$y$	40	54	68

Contexts and independent checks

Problem 6 · Feed cost
days $x$ 0 3 6 9
cost $y$ (€) 0 12 24 36
1. Calculate the rate with units.
2. Interpret the rate in a sentence.
3. Write an equation for cost after $x$ days.
changes → units → meaning → equation →
Problem 7 · Choose a sponsor drive
Two local riding clubs offer fundraising plans. The ordered pairs show (days, total donations in euros). Plan A: $(0,10),(2,18)$. Plan B: $(0,4),(3,19)$.
1. Calculate Plan A's donation rate in euros/day.
2. Calculate Plan B's donation rate in euros/day.
3. Choose the plan with the greater daily rate and state the difference.
Decision: Mia will carry the faster fundraising rate into the building proposal in Lesson 27.
rate A → rate B → compare →
Problem 8 · Uneven x-steps
$x$ 1 4 10
$y$ 7 19 43
1. Calculate the rate from the first pair of columns.
2. Calculate the rate from the second pair.
3. Choose CONSTANT or NOT CONSTANT.
divide matching changes → compare rates →
Problem 9 · Missing-coordinate review
A line with slope $3$ passes through $(2,5)$ and $(8,y)$.
1. Calculate $\Delta x$.
2. Calculate $\Delta y$.
3. Calculate $y$.
Δx → mΔx → ending y →
Problem 10 · Inequality retention
A water container starts with $50$ L and loses $6$ L per hour. Let $h=$ hours.
1. Write an inequality for when at most $14$ L remain.
2. Solve and interpret the inequality.
model → negative coefficient → reverse sign →
Problem 11 · Decreasing arithmetic retention
A training plan uses $72,65,58,51,\ldots$ jumps. The values decrease.
1. Calculate the signed common difference using next term minus previous term.
2. Write an explicit formula in the form $a_n=a_1+(n-1)d$, keeping negative $d$ in parentheses.
3. Calculate $a_{10}$.
next − previous → signed formula → substitute →

days $x$	0	3	6	9
cost $y$ (€)	0	12	24	36

$x$	1	4	10
$y$	7	19	43

Show answers

Problem 1a

$\Delta x=2,2,2$.

Problem 1b

$\Delta y=6,6,6$.

Problem 1c

Rate $=6\div2=3$.

Problem 1d

CONSTANT.

Problem 2a

DECREASES.

Problem 2b

Rate $=-8$ percentage points/hour.

Problem 2c

Battery level decreases by $8$ percentage points per hour.

Problem 3a

The changes are $+3,+4,+5$.

Problem 3b

NOT CONSTANT.

Problem 3c

Equal $x$-steps produce different $y$-changes.

Problem 4a

Using $(1,8)$ and $(5,20)$: $m=12\div4=3$.

Problem 4b

From $x=1$ to $x=3$, $y$ rises by $3\times2=6$, so the missing value is $14$.

Problem 4c

$(14-8)\div(3-1)=3$ and $(20-14)\div(5-3)=3$.

Problem 5a

$m=(54-40)\div(2-0)=7$.

Problem 5b

$b=40$.

Problem 5c

$y=7x+40$.

Problem 6a

$m=12\div3=€4$/day.

Problem 6b

Feed cost increases by €4 per day.

Problem 6c

$y=4x$.

Problem 7a

Plan A: $(18-10)\div2=4$.

Problem 7b

Plan B: $(19-4)\div3=5$.

Problem 7c

Plan B is greater by $1$ y-unit per x-unit.

Problem 8a

$(19-7)\div(4-1)=12\div3=4$.

Problem 8b

$(43-19)\div(10-4)=24\div6=4$.

Problem 8c

CONSTANT.

Problem 9a

$\Delta x=8-2=6$.

Problem 9b

$\Delta y=3\times6=18$.

Problem 9c

$y=5+18=23$.

Problem 10a

$50-6h\leq14$.

Problem 10b

$$50-6h\leq14\quad\mid-50$$$$-6h\leq-36\quad\mid\div(-6)$$$$h\geq6$$At most $14$ L remain after at least $6$ hours.

Problem 11a

$d=65-72=-7$.

Problem 11b

$a_n=72+(n-1)(-7)$.

Problem 11c

$a_{10}=72+9(-7)=9$.

Chapter 2 → Lesson 27: The blueprint meetingThe ledger audit gives Mia evidence she can defend: which patterns are dependable, how fast feed costs grow, and which sponsor plan raises money faster. Next, those numbers stop being entries in a book—they become choices in a real stable design and budget.

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