Corrected exercises: index numbers, rates and constant dollars (360-300)

$\frac{1{,}410-1{,}250}{1{,}250}=\frac{160}{1{,}250}=0.128$, an increase of $12.8\%$.

$\frac{3{,}330-3{,}600}{3{,}600}=\frac{-270}{3{,}600}=-0.075$, a decrease of $7.5\%$. Pitfall: dividing by the final value, $\frac{270}{3{,}330}\approx 8.1\%$.

In points: $18.6-15.0=3.6$ percentage points. In percent: $\frac{3.6}{15.0}=0.24$, an increase of $24.0\%$. Pitfall: announcing an increase "of $3.6\%$", which confuses the difference of two rates with their relative change.

The multiplier is $1.065$: $2{,}800\times 1.065=2{,}982$ dollars.

A decrease of $12\%$ is the multiplier $0.88$, so $\text{start}=\frac{1{,}980}{0.88}=2{,}250$ dollars. Pitfall: adding $12\%$ to the final value, $1{,}980\times 1.12\approx 2{,}218$, which measures the decrease against the wrong value.

$1.25\times 0.84=1.05$: the overall multiplier is $1.05$, an increase of $5.0\%$. Pitfall: adding the rates, $25-16=9\%$.

$I=\frac{21{,}160}{18{,}400}\times 100=1.15\times 100=115.0$: the quantity has increased by $15\%$ since 2019.

Divide each index number by the one of 2020, then multiply by $100$. In 2024: $\frac{140}{112}\times 100=125.0$. In 2015: $\frac{100}{112}\times 100\approx 89.3$. Pitfall: subtracting $12$ points from each index number, which gives $128$ and $88$ and does not preserve the ratios.

$54{,}000\times\frac{120}{160}=54{,}000\times 0.75=40{,}500$ dollars of 2012. Pitfall: multiplying by $\frac{160}{120}$, which gives $72{,}000$; since prices have risen, an amount expressed in dollars of the earlier year must be SMALLER.

We look for $t$ such that $5{,}000\times(1+t)^{4}=6{,}400$, so $(1+t)^{4}=1.28$ and $1+t=1.28^{\frac{1}{4}}\approx 1.0637$: about $6.4\%$ per year. Pitfall: dividing the total increase by $4$, $\frac{28}{4}=7\%$, which applied four times gives $1.07^{4}\approx 1.311$ and not $1.28$.

In percentage POINTS: $7.5-6.0=1.5$ points. In PERCENT: $\dfrac{7.5-6.0}{6.0}=0.25$, an increase of 25%. The two numbers, 1.5 and 25, describe the same event and are not interchangeable. The writing rule is simple: a difference of two percentages is said in POINTS; a ratio of two percentages is said in PERCENT.

First sentence: "the party gains 4 POINTS, going from 32% to 36%". Second sentence: "voting intentions for the party rise by 12.5%, since $\dfrac{36-32}{32}=0.125$". The first is the one political analysts use, because it compares directly with the margin of error, which is also expressed in points. The second is correct but misleading in this context: it suggests spectacular progress where four points may fit within the sampling noise.

Relative change: $\dfrac{782-850}{850}=\dfrac{-68}{850}=-0.08$, a decrease of 8.0%. To get back to 850 from 782: $\dfrac{850-782}{782}=\dfrac{68}{782}\approx 0.0870$, an increase of 8.7%. So it takes a LARGER increase than the decrease suffered to get back to the starting point.

Because a relative change is always computed on the STARTING value, and the two movements do not have the same starting value: 850 for the decrease, 782 for the increase. The same absolute gap of 68 divided by a smaller denominator gives a larger percentage. Formally, a decrease of $t$ followed by an increase of $t$ gives the factor $\left(1-t\right)\left(1+t\right)=1-t^{2}<1$: you never return to the starting point. To undo a decrease of $t$, you need an increase of $\dfrac{t}{1-t}$.

It is correct, and that is precisely what makes it tricky. Going from 10% to 12% is indeed a relative increase of $\dfrac{12-10}{10}=20\%$, and it is also an increase of 2 POINTS. The wording chosen, though not false, is the one with the most impact, and the hurried reader will remember that a fifth of the population has fallen into poverty. A rigorous newspaper writes both: "the rate goes from 10% to 12%, two points higher, an increase of 20%".

Multipliers: $1.10$ then $0.90$, so $1.10\times 0.90=0.99$. The price has therefore FALLEN by 1%. An item at 200 dollars goes to 220, then to 198. The order of the two changes makes no difference, since multiplication is commutative: a good argument to convince a student that the result does not depend on the direction of reading.

Overall multiplier: $1.03^{5}\approx 1.1593$, an increase of 15.93%. The naive addition would give 15%, which underestimates by nearly a full point. The gap comes from COMPOUND INTEREST: the increase of the third year applies to an amount that has already risen twice. The gap grows quickly with the number of periods: over 20 years, $1.03^{20}\approx 1.806$ against 1.60 for the addition, a difference of 20 points.

Multiplier: $0.98^{12}\approx 0.7847$. About 78.5% of the population therefore remains, a loss of 21.5% in twelve years, whereas naive addition would announce 24%. Here the naive error EXAGGERATES the loss, because each decrease applies to a population that is already reduced. Remember the rule: compounding softens decreases and amplifies increases compared with addition.

We look for $t$ such that $42{,}000\times \left(1+t\right)^{7}=51{,}500$, so $\left(1+t\right)^{7}=\dfrac{51{,}500}{42{,}000}\approx 1.22619$ and $1+t=1.22619^{1/7}\approx 1.02958$. The average annual rate is therefore about 2.96%. It is a GEOMETRIC mean and not an arithmetic one: dividing the total increase of 22.6% by 7 would give 3.23%, a number that does not rebuild the final result if you apply it seven times.

No. The two discounts apply in succession: $0.70\times 0.80=0.56$, so the customer pays 56% of the price and the total discount is 44%, not 50%. On a 100-dollar item: 70 dollars after the first discount, then 56 dollars after the second. The gap of six points is exactly the product $0.30\times 0.20=0.06$, that is, the discount that would be counted twice if you added. It is the commercial version of the same pitfall, and it is perfectly legal because the ad never says that the discounts add up.

$I=\dfrac{1{,}186}{780}\times 100\approx 152.1$. Rent has therefore risen by about 52% in six years. An index number reads directly as a percentage of the base year, which is its whole point: 152 means "152% of what it was", so 52% more.

$I=\dfrac{40{,}000}{34{,}200}\times 100\approx 117.0$, an increase of 17%. Both series start at 100 and end at 152 and 117: on the graph, the gap widens steadily, which no comparison of raw dollars would have shown, since 1,186 dollars and 40,000 dollars cannot be compared.

Ratio of the index numbers: $\dfrac{152.1}{117.0}\approx 1.300$. Rent has therefore risen 30% faster than income. In other words, the share of income spent on rent has been multiplied by 1.30: if it was 27% at the start, it is now about 35%. It is this quotient of index numbers, and not the difference 152 minus 117, that has a meaning: subtracting two index numbers would give 35 "index points", a quantity whose interpretation depends on the base chosen.

Changing the base means dividing all the index numbers by the one of the new reference year and multiplying by 100. Year 0: $\dfrac{100}{129}\times 100\approx 77.5$. Year 5: $\dfrac{152.1}{129}\times 100\approx 117.9$. We check that year 3 is now exactly 100. The TREND is unchanged: the ratio of year 5 to year 0 is $\dfrac{117.9}{77.5}\approx 1.521$ in the new base as in the old one. An index number therefore contains no absolute information, only ratios.

Yes, as soon as the value is below that of the base year. An index number of 87 means that the quantity is now worth only 87% of its reference value, a decrease of 13%. This is a common reading for indexes of industrial production or school enrolment. Be careful not to say "the index fell by 13 points" if the base is not 100: the correct wording always compares to the base explicitly named, for example "87, base 100 in 2015".

Conversion: $68{,}000\times \dfrac{100}{150}=68{,}000\times 0.6667\approx 45{,}333$ dollars of 2005. The rule is to multiply by the ratio "index of the TARGET year over index of the year of the amount". You can check the direction of the calculation with common sense: since prices have risen, a 2025 dollar is worth less, so the converted amount must be SMALLER than 68,000.

Purchasing power has FALLEN: 45,333 dollars of 2005 against 48,000 in 2005, a decrease of $\dfrac{45{,}333-48{,}000}{48{,}000}\approx -0.0556$, about 5.6%. This is the central result of the chapter: a salary that rises by 42% in twenty years can mean impoverishment, because prices rose by 50%.

You would have needed $48{,}000\times \dfrac{150}{100}=72{,}000$ dollars in 2025. The gap with the 68,000 actually paid, that is 4,000 dollars, measures exactly the loss of purchasing power in 2025 dollars. We can check consistency: $\dfrac{68{,}000}{72{,}000}\approx 0.944$, which indeed gives back the decrease of 5.6% from the previous question.

We look for $t$ such that $\left(1+t\right)^{20}=\dfrac{150}{100}=1.5$, so $1+t=1.5^{1/20}\approx 1.02048$ and $t\approx 2.05\%$ per year. An annual inflation rate of 2%, which looks harmless, is therefore enough to cost a nominal amount a third of its purchasing power in twenty years, since $\dfrac{1}{1.5}\approx 0.667$.

"Your salary rose by 42%, prices by 50%: you are 5.6% poorer." The calculation fits on one line: $\dfrac{1.4167}{1.50}\approx 0.944$, starting from the exact multiplier of the salary, $\dfrac{68{,}000}{48{,}000}=1.4167$, and not from its rounding to $1.42$. This is the general formula for purchasing power, multiplier of the salary divided by multiplier of prices, and it is the only calculation to keep from this exercise. Any salary whose increase is smaller than inflation represents a real decrease, whatever the headline figure.

The LABOUR FORCE is $384{,}000+32{,}000=416{,}000$ people, those who have a job or are actively looking for one. Unemployment rate: $\dfrac{32{,}000}{416{,}000}\approx 0.0769$, that is 7.7%. Participation rate: $\dfrac{416{,}000}{640{,}000}=0.65$, that is 65%. The denominator of the unemployment rate deserves emphasis: it is not the total population, it is the labour force, which excludes full-time students, retirees and people who are not looking.

Counting the 18,000 discouraged people as unemployed, the labour force becomes $416{,}000+18{,}000=434{,}000$ and the number of unemployed $32{,}000+18{,}000=50{,}000$. The rate becomes $\dfrac{50{,}000}{434{,}000}\approx 0.1152$, that is 11.5%. A gap of nearly four points for one and the same economic reality, only because the definition of "looking for work" was changed. This is why Statistics Canada publishes several complementary rates, R1 to R8, rather than a single one.

The labour force falls to $416{,}000-12{,}000=404{,}000$ and the unemployed to $20{,}000$. The rate becomes $\dfrac{20{,}000}{404{,}000}\approx 0.0495$, that is 5.0% instead of 7.7%. The rate has therefore DROPPED by 2.7 points although no job has been created: people simply stopped looking. This is the exact mechanism of discouragement, and it operates strongly at the end of a recession.

Because the rate can fall for two opposite reasons: because the numerator falls, unemployed people find a job, which is good news; or because the denominator falls, unemployed people leave the labour force, which is bad news. Both movements give the same change in the rate. To decide, you have to look at the EMPLOYMENT RATE and the participation rate: if the unemployment rate falls and the employment rate falls too, it is discouragement.

Employment rate: $\dfrac{384{,}000}{640{,}000}=0.60$, that is 60%. It complements the unemployment rate because its denominator, the total population aged 15 and over, does not depend on job-search behaviour: it is NOT AFFECTED by discouragement. When unemployed people withdraw, the unemployment rate falls and the employment rate stays the same, which immediately reveals the unintended trick. Publishing both together is the correct practice, and it is what statistical agencies do.

Crude rate of A: $0.60\times 1.0+0.30\times 6.0+0.10\times 50.0=0.6+1.8+5.0=7.4$ per 1,000. Crude rate of B: $0.40\times 1.2+0.30\times 7.0+0.30\times 55.0=0.48+2.1+16.5=19.08$ per 1,000. Region B therefore shows a crude rate two and a half times higher.

At equal age, region B is indeed the deadlier, but only slightly: 1.2 against 1.0 among those under 40, 7.0 against 6.0 among the 40 to 64, 55.0 against 50.0 among seniors. The ratios are 1.20, 1.17 and 1.10 respectively, an excess mortality of about 10 to 20% in each bracket, nowhere near the ratio of 2.58 between the crude rates.

Standardized rate of A: $0.50\times 1.0+0.30\times 6.0+0.20\times 50.0=0.5+1.8+10.0=12.3$. Standardized rate of B: $0.50\times 1.2+0.30\times 7.0+0.20\times 55.0=0.6+2.1+11.0=13.7$. We apply the ACTUAL specific rates of each region to a COMMON, fictitious age structure: the two numbers obtained describe what would happen if the two regions had the same age pyramid.

Crude gap: $19.08-7.4=11.68$ per 1,000, with a ratio of 2.58. Standardized gap: $13.7-12.3=1.4$, with a ratio of 1.11. Standardization removed the effect of the AGE STRUCTURE: it shows that more than 88% of the crude gap came from the fact that region B has three times as many seniors, and not from a difference in mortality at equal age. This is the typical result of this tool, and it almost always overturns the naive interpretation.

The mayor is comparing crude rates, so mainly age pyramids. Correction: "our population is younger, which explains most of our lower crude rate. With a comparable age structure, our mortality is 12.3 per 1,000 against 13.7 for our neighbour, about 10% lower, which remains a real advantage but a much more modest one." It should be added that this residual advantage itself needs an explanation, income, access to care, environment, and that a standardized rate is never more than a starting point for analysis.

Exact: we solve $\left(1.02\right)^{n}=2$, so $n=\dfrac{\ln 2}{\ln 1.02}=\dfrac{0.6931}{0.019803}\approx 35.0$ years. Rule of 70: $\dfrac{70}{2.0}=35$ years. The agreement is excellent, and it stays so for all the usual rates, say from 0.5% to 10%.

Doubling requires $\left(1+t\right)^{n}=2$, so $n\ln\left(1+t\right)=\ln 2\approx 0.693$. For small $t$, $\ln\left(1+t\right)\approx t$, hence $n\approx \dfrac{0.693}{t}$. Expressing the rate as a PERCENTAGE, $t=\dfrac{p}{100}$, we get $n\approx \dfrac{69.3}{p}$, rounded to 70 because that number has many whole divisors, which makes mental arithmetic easy: 70 divided by 2, by 5, by 7 or by 10 comes out even.

$180{,}000\times \left(1.014\right)^{25}=180{,}000\times 1.4159\approx 254{,}800$ inhabitants. You can check with the rule of 70: the doubling time is $\dfrac{70}{1.4}=50$ years, so in 25 years, half a period, the population should be multiplied by about $\sqrt{2}\approx 1.414$, which agrees with the exact calculation to within two thousandths.

Tripling at 3%: $n=\dfrac{\ln 3}{\ln 1.03}=\dfrac{1.0986}{0.029559}\approx 37.2$ years. Meanwhile, the country at 0.5% is multiplied by $\left(1.005\right)^{37.2}\approx 1.204$, an increase of 20%. In under forty years, a gap of 2.5 points in the annual rate therefore produces a ratio of $\dfrac{3}{1.204}\approx 2.5$ between the two countries. It is this mechanism, and not any particular event, that explains the divergence of demographic and economic trajectories over half a century.

"At 3% per year, everything doubles in 23 years and is multiplied by 8 in 70 years." The calculation: $\dfrac{70}{3}\approx 23.3$ years per doubling, and 70 years contain three doublings, hence a factor $2^{3}=8$. This sentence is enough to make clear that no constant growth rate is modest over a long period, and it is the central argument of every debate on the growth of finite resources.

FALSE, or at least ambiguous to the point of being false. The decrease is 2 percentage POINTS, and in percent it is $\dfrac{6-8}{8}=-25\%$. Writing "fell by 2%" suggests the second reading and gives a number twelve times too small. Correct statement: the unemployment rate fell by two points, from 8% to 6%, a relative decrease of 25%.

FALSE. Rates compound: $1.04^{10}\approx 1.4802$, an increase of 48.0% and not 40%. The gap of eight points comes from the fact that each increase applies to an amount that has already risen. Correct statement: at 4% per year, tuition rises by about 48% in ten years, and adding the rates is only a valid approximation over two or three periods.

FALSE, the prices are missing. The nominal increase is $\dfrac{60{,}000}{45{,}000}=1.333$, that is 33%; but if the price index has risen by 40% over the period, purchasing power is $\dfrac{1.333}{1.40}\approx 0.952$, a LOSS of 4.8%. Correct statement: the nominal salary rose by 33%, and purchasing power can only be computed by dividing this multiplier by that of the price index.

FALSE if we speak in percent, true if we speak in index points. The index gained 30 INDEX POINTS, which corresponds to an increase of $\dfrac{150-120}{120}=0.25$, that is 25%. The confusion comes from the fact that an index number looks like a percentage without being one. Correct statement: the index goes from 120 to 150, which represents a price increase of 25% relative to the previous period, and of 50% relative to the base year.

FALSE twice over. First, a crude rate depends mostly on the age structure: region A is probably younger, and standardization would bring the gap down to almost nothing. Second, a death rate is not the inverse of a lifespan: life expectancy is computed from the age-specific rates in a life table, and a crude rate half as large never corresponds to a doubled life expectancy. Correct statement: crude rates cannot be compared between populations with different structures; they must be standardized, and to speak about lifespan you have to go through life expectancy.

NOMINAL increase: $\dfrac{68{,}000-48{,}000}{48{,}000}=\dfrac{20{,}000}{48{,}000}\approx 0.4167$, that is 41.7%. REAL increase: multiplier of the salary divided by multiplier of prices, $\dfrac{1.4167}{1.50}\approx 0.9444$, a 5.6% DECREASE in purchasing power. Both figures are exact and tell opposite stories: that is exactly why the two sides of a negotiation each quote their own.

Full catch-up: $48{,}000\times 1.50=72{,}000$ dollars. From 68,000, this represents $\dfrac{72{,}000-68{,}000}{68{,}000}\approx 0.0588$, an increase of 5.9%. Note the asymmetry already met: the loss is 5.6% and the catch-up requires 5.9%, because the two percentages are not computed on the same base.

Multiplier of the salary over three years: $1.03^{3}\approx 1.0927$. Multiplier of prices: $1.022^{3}\approx 1.0675$. Purchasing power: $\dfrac{1.0927}{1.0675}\approx 1.0236$, a gain of 2.4% over three years. Purchasing power therefore rises, but it does not make up the loss: starting from 0.9444, we reach $0.9444\times 1.0236\approx 0.9667$, still 3.3% below the 2005 level.

Each year, purchasing power is multiplied by $\dfrac{1.03}{1.022}\approx 1.00783$. We must solve $\left(1.00783\right)^{n}=\dfrac{1}{0.9444}=1.0588$, so $n=\dfrac{\ln 1.0588}{\ln 1.00783}\approx \dfrac{0.05716}{0.007800}\approx 7.3$ years. It would therefore take more than seven years at this pace to return to the purchasing power of 2005, assuming inflation stays at 2.2%, which is the most fragile assumption of the whole calculation.

Check: $1.03^{3}=1.092727$, that is 9.27%, rounded to 9.3%. The figure is therefore exact. The word "cumulative" is technically correct, since it does designate the compounded effect of the three increases, and it is shrewd in negotiation, because it is the largest of the available figures to describe the same offer: one could have said 3% per year, or 9.0% by simple addition, or 2.4% real gain, or even a residual loss of 3.3% relative to 2005. Five exact numbers for a single offer: the lesson of the chapter is that a news release is judged by the choice of indicator, not by the accuracy of the calculation.

No, prices CONTINUE TO RISE, just more slowly: inflation of 2.4% means that the general price level is 2.4% higher than a year earlier. The move from 3.1% to 2.4% is called DISINFLATION, that is, a slowdown in the rise. DEFLATION is an actual fall in prices, hence a NEGATIVE rate, and it is a rare and dreaded phenomenon, because it encourages people to postpone purchases. An index that goes from 130 to 134.0 and then to 137.2 illustrates inflation and disinflation at once: it keeps rising, by 3.1% and then by only 2.4%.

Wages up 4.0%, prices up 2.4%: purchasing power is multiplied by $\dfrac{1.040}{1.024}\approx 1.0156$, a gain of 1.6%. The naive subtraction would also give 1.6 points, which is here a good approximation because the rates are small; the gap between the two methods only appears in the third decimal. You should still keep the reflex of the quotient, because with two-digit rates the approximation becomes plainly wrong.

Gross domestic product per capita is the ratio of total product to population: $\dfrac{1+g}{1.018}=1.012$, so $1+g=1.012\times 1.018\approx 1.03022$ and total growth is about 3.0%. This kind of conversion is crucial in public debate: total growth of 3% in a country whose population grows by 1.8% is only a gain of 1.2% per person, and the same growth in a country with a stable population would be 3%.

The SERIES is missing. "Lowest level in two years" says neither where we are coming from, nor whether the decline is steady or erratic, nor above all whether the participation rate has moved: a fall in unemployment accompanied by a fall in the employment rate would signal discouragement rather than an improvement. The margin of error of the labour force survey is also missing, of the order of two tenths of a point, which can by itself explain a drop from 5.9% to 5.8%. Finally, the two-year window is a choice: over ten years, the same 5.8% might be a high level.

"In March, prices are still 2.4% higher than a year earlier, against 3.1% in February: the rise is slowing but not reversing. With wages rising by 4.0%, purchasing power gains about 1.6% over one year. Total gross domestic product grows by about 3.0%, but with the population growing by 1.8%, the gain per person is limited to 1.2%. The unemployment rate falls by a tenth of a point to 5.8%, a movement of the same order as the margin of error of the survey, which will need to be confirmed by the trend in the employment rate." Three ideas: name what rises and what slows, bring each rate back to its base, and point out what cannot be distinguished from noise.