Lesson 20a · Repair workshop

Trail Steps & Money Seeds

Two small ideas to make solid: signed steps in sequences and percentages in simple interest.

This is a repair stop, not a repeat of the review. First we will make each idea visible. Then you will use a short path every time, so a minus sign or percent sign cannot quietly disappear.

Repair 1 · A sequence walking downhill

Direction check: the terms get smaller, so the common difference must be negative. Always calculate it in this order: next term minus previous term.

$$d=65-72=-7$$

Worked example · keep the negative sign attached

A riding drill has $a_1=48$ seconds and improves by $3$ seconds per round, so $d=-3$.

Start: $a_1=48$Step: $d=-3$Formula: $a_n=a_1+(n-1)d$Substitute: $48+(n-1)(-3)$

For round $6$: $a_6=48+5(-3)=33$ seconds. There are five steps from term 1 to term 6—not six.

Problem 1 · Find the signed step
A sketch challenge uses $90,82,74,66,\ldots$ seconds.
Calculate the common difference using $d=\text{next term}-\text{previous term}$. Include its sign.
next − previous → direction check →
Problem 2 · Build the formula
A horse completes $64,59,54,49,\ldots$ metres in successive training intervals.
Write the explicit formula in the form $a_n=a_1+(n-1)d$. Keep the negative difference in parentheses.
name $a_1$ → calculate $d$ → formula →
Problem 3 · Use the formula
For the sequence in Problem 2, calculate $a_8$. State the value with the unit metres.
$a_8=64+(8-1)(-5)$ →

Repair 2 · A percent becomes a decimal

$3$ shaded squares out of $100$: $3\%=\frac{3}{100}=0.03$

Worked example · four boxes, in order

A savings account holds €600 at $4\%$ simple interest for $2$ years.

Percent: $4\%$Decimal: $r=0.04$Interest: $I=600\times0.04\times2=48$Total: $A=600+48=648$

The interest is €48. The total amount is €648. Those are different quantities.

Say the path aloud: percent to decimal; substitute in $I=Prt$; calculate interest; add principal only if the question asks for the total.

Problem 4 · Open the percent gate
A bank offers $2.5\%$ per year.
Convert $2.5\%$ to a decimal rate $r$. Write both the division by $100$ and the decimal.
$2.5\div100$ → decimal →
Problem 5 · Name the formula inputs
Mia deposits €800 at $3\%$ simple interest per year for $2.5$ years.
State $P$, decimal $r$, and $t$, each with its meaning or unit. Do not calculate the interest yet.
principal → decimal rate → time →
Problem 6 · Calculate interest
Use the values from Problem 5.
Calculate the interest $I$ using $I=P\times r\times t$. Label the answer interest and include euros.
substitute → multiply → label →
Problem 7 · Separate interest from total
Use the €800 principal and your interest from Problem 6.
Calculate the total amount $A=P+I$. Label the answer total amount and include euros.
principal + interest → label →

Show answers only after all seven problems

Problem 1

$d=82-90=-8$. The negative sign matches the decreasing terms.

Problem 2

$a_1=64$, $d=59-64=-5$, so $a_n=64+(n-1)(-5)$.

Problem 3

$a_8=64+7(-5)=29$ metres.

Problem 4

$2.5\div100=0.025$, so $r=0.025$.

Problem 5

$P=€800$ principal, $r=0.03$ per year, and $t=2.5$ years.

Problem 6

$I=800\times0.03\times2.5=€60$ interest.

Problem 7

$A=P+I=€800+€60=€860$ total amount.

Back to the expedition trailLater lessons will revisit these ideas in new settings. When a sequence goes down, carry the negative sign. When money earns a percent, pass through the decimal gate before using the formula.

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