Ratios, Proportions & Percents — 700-800

The 700-800 Patterns: Ratios, Proportions & Percents

Part 1 of 3 — Multipliers, Chains, and Working Backward

At this level, percent questions are multiplier questions. The instant you translate "$15\%$ off" into $\times\, 0.85$ and "$8\%$ tax" into $\times\, 1.08$, the hard tier collapses into arithmetic.

Archetype 1: Successive Percent Change (Multipliers Compose)

Changes multiply — they never add. Up $25\%$ then down $20\%$ is $1.25 \times 0.80 = 1.00$: no net change. Down $15\%$ then down $8\%$ more is $0.85 \times 0.92 = 0.782$, a $21.8\%$ total drop — not $23\%$.

Worked example. A price rises $30\%$, then falls $30\%$. Net change? $1.30 \times 0.70 = 0.91$ — down $9\%$. The "back where it started" instinct is the planted answer.

Archetype 2: Reverse Percent (Divide by the Multiplier)

Given the FINAL value, recover the original by dividing by the full multiplier chain — never by applying the percent "backward."

Worked example. Final price \125.12$after$15%$off then$8%$off: original$= 125.12 \div (0.85 \times 0.92) = 125.12 \div 0.782 = $160$. Multiplying$125.12$by$1.15$or$1.23$gives the two standard wrong answers, because a$15%$decrease is NOT undone by a$15%$ increase — the base changed.

Archetype 3: Percent of a Percent

"$60\%$ of students take a language; $35\%$ of THOSE take Spanish" chains to $0.60 \times 0.35 = 0.21$ of the whole school. Given the final count, divide by the combined rate: $189 \div 0.21 = 900$ students. The one-step divisions ($189 \div 0.35$, $189 \div 0.60$) are both planted.

Archetype 4: Unit-Rate Chains

Hard ratio items chain three or more conversions (mg per kg per dose; kg $\rightarrow$ cost; km $\rightarrow$ mi $\rightarrow$ dollars). Write the chain as one line with units, cancel as you go, and the units of the answer tell you when to stop. Every intermediate value in the chain — in its wrong units — appears in the options.

Traps & Speed: Ratios, Proportions & Percents

Part 2 of 3 — Every Distractor Is a Unit or a Direction

The Distractor Recipe on Percent/Ratio Items

1. The intermediate in the wrong units. Grams when bags were asked. Euros when dollars were asked. Kilograms when the ask was a COST. The number is correct — the units aren't. A quantity masquerading as a price is the hard tier's favorite move.
2. The wrong direction. Multiplying by $1.06$ when reversing a tax (should divide); dividing by $1.6$ when the conversion needed multiplying. Sanity-check with size: if a euro is worth MORE than a dollar, the dollar figure must be BIGGER.
3. Added percents. $15\%$ then $8\%$ is never $23\%$. Any option built from the summed percent is wrong on principle — you can eliminate it without computing.
4. The dropped step. One factor of the chain skipped: the per-day answer on a per-week ask, the full-hectare answer on a half-hectare field.

Speed Techniques

• Write the multiplier chain before any arithmetic: "paid $= P \times 0.80 \times 1.06$" turns every forward/backward question into one Desmos division. Type $89.04 / (0.8 \times 1.06)$ — done.
• Reverse = divide. Original $=$ final $\div$ (product of multipliers). No exceptions, no "add the percent back."
• Unit-cancel on one line: $396 \text{ lb} \div 2.2 \rightarrow \text{kg} \times 0.85 \rightarrow \text{euros} \times 1.10 \rightarrow \text{dollars}$. When the units read as the asked-for units, stop; if an option matches a mid-chain value, that option is the trap, and its presence CONFIRMS your chain.
• Elimination first: on reverse-percent items, the original must be BIGGER than the final after a net discount. Cross out every option smaller than the final price before computing.

Ratios, Proportions & Percents: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Pace yourself at about 90 seconds per question. Hard ratio and percent items are rarely hard arithmetic; they are long chains where the finish line is one link past where the work feels done.

Run the same routine on each:

1. Fifteen seconds — build the chain. Write the sequence of conversions with units attached, or the multiplier product ($\times\, 1.60$, then $\times\, 0.70$), before computing anything.
2. Fifty seconds — execute in one calculator expression rather than in separate steps, so no intermediate ever gets mistaken for the answer.
3. Twenty-five seconds — audit the ask. Does the stem want a cost or a quantity? A percent or a dollar amount? A total or a difference? A per-day figure or a per-month one?

One extra guard, specific to this topic: squared scale factors. Whenever a drawing, map, or model is scaled by a linear factor $k$, its areas scale by $k^{2}$. That single fact decides one of the four questions below.