Math worksheet with graphing and algebra problems.

Let’s solve each problem step by step.

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Problem 1: Science Exam Study Hours vs Final Exam Score

Data:
Hours (x): 0, 1, 2, 3, 4, 5
Score (y): 60, 70, 80, 90, 95, 98

a) Scatterplot — we’ll skip drawing since this is text-based, but imagine plotting points: (0,60), (1,70), (2,80), (3,90), (4,95), (5,98)

b) Equation for line of best fit?

We can estimate the line using two points that seem to lie on a straight trend. Let’s pick (0,60) and (3,90).

Slope = (90 - 60)/(3 - 0) = 30/3 = 10

So equation: y = 10x + 60

Check with another point: x=2 → y=80 ✔️
x=4 → y=100? But actual is 95 → close
x=5 → y=110? Actual is 98 → not perfect, but reasonable for estimation.

Alternatively, use linear regression formula if needed — but for middle/high school level, picking two good points is acceptable.

But let’s do proper least squares to be accurate.

n = 6
Σx = 0+1+2+3+4+5 = 15
Σy = 60+70+80+90+95+98 = 493
Σxy = (0×60)+(1×70)+(2×80)+(3×90)+(4×95)+(5×98) = 0 + 70 + 160 + 270 + 380 + 490 = 1370
Σx² = 0+1+4+9+16+25 = 55

Slope m = [nΣxy - ΣxΣy] / [nΣx² - (Σx)²]
= [6×1370 - 15×493] / [6×55 - 225]
= [8220 - 7395] / [330 - 225]
= 825 / 105 ≈ 7.857

Intercept b = [Σy - mΣx]/n = [493 - 7.857×15]/6 = [493 - 117.855]/6 = 375.145/6 ≈ 62.524

So equation: y ≈ 7.86x + 62.52

(Rounded to two decimals)

c) Predict score for 6 hours:

y = 7.86×6 + 62.52 = 47.16 + 62.52 = 109.68

But wait — max possible exam score is usually 100. So prediction >100 may not make sense in real world, but mathematically it’s what the line says.

d) Can line go on forever? No — because scores can’t exceed 100 (probably), and studying more than 6 hours might not keep increasing score linearly — diminishing returns. Also, negative hours don’t make sense.

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Problem 2: Height vs Shoe Size

Data:
Height (in): 64, 65, 68, 70, 72, 75, 80
Shoe size: 7, 8, 9, 10, 11, 12, 13

Wait — shoe sizes are listed as: 7, 8, 9, 10, 11, 12, 13? That seems too high for men’s US sizes — maybe typo? Or perhaps it's European? But problem says “randomly selected men” — likely US sizes. Max men’s US size is around 15, so 13 is possible but rare. Anyway, proceed with given data.

Actually looking back at image description — user didn't provide full table clearly. From context:

Assume:

Height: 64, 65, 68, 70, 72, 75, 80
Shoe size: 7, 8, 9, 10, 11, 12, 13 → this looks like an exact linear pattern!

Check slope between first and last:

(13 - 7)/(80 - 64) = 6/16 = 0.375

Equation: y - 7 = 0.375(x - 64)
→ y = 0.375x - 24 + 7 = 0.375x - 17

Check x=65: 0.375*65 = 24.375 -17 = 7.375 → should be 8? Not matching.

Wait — maybe shoe sizes are: 7, 8, 9, 10, 11, 12, 13 corresponding to heights 64 to 80? Then difference per inch: from 64 to 80 is 16 inches, shoe size increases 6 units → 6/16 = 0.375 per inch.

But at x=64, y=7 → intercept: 7 = 0.375*64 + b → 7 = 24 + b → b = -17

Then at x=65: y=0.375*65 -17 = 24.375 -17 = 7.375 ≠ 8 → inconsistency.

Perhaps the shoe sizes are misread? Maybe they are: 7, 8, 9, 10, 11, 12, 13 but assigned incorrectly?

Alternative approach: assume perfect correlation for simplicity (as often done in such problems unless specified otherwise).

Notice: height increases by 1 or 2 or 3 inches, shoe size increases by 1 each time? Let’s list:

If we pair:

64 → 7
65 → 8
68 → 9 ← jump of 3 inches for 1 size
70 → 10 ← 2 inches
72 → 11 ← 2 inches
75 → 12 ← 3 inches
80 → 13 ← 5 inches

Not consistent.

Better to calculate regression.

n=7
Σx = 64+65+68+70+72+75+80 = let’s compute:
64+65=129; +68=197; +70=267; +72=339; +75=414; +80=494
Σy = 7+8+9+10+11+12+13 = 70
Σxy = 64*7 + 65*8 + 68*9 + 70*10 + 72*11 + 75*12 + 80*13
Calculate:

64*7 = 448
65*8 = 520
68*9 = 612
70*10 = 700
72*11 = 792
75*12 = 900
80*13 = 1040
Sum: 448+520=968; +612=1580; +700=2280; +792=3072; +900=3972; +1040=5012

Σx² = 64²=4096, 65²=4225, 68²=4624, 70²=4900, 72²=5184, 75²=5625, 80²=6400
Sum: 4096+4225=8321; +4624=12945; +4900=17845; +5184=23029; +5625=28654; +6400=35054

Now slope m = [nΣxy - ΣxΣy] / [nΣx² - (Σx)^2]
= [7*5012 - 494*70] / [7*35054 - 494^2]

Compute numerator: 7*5012 = 35084; 494*70 = 34580; diff = 35084 - 34580 = 504

Denominator: 7*35054 = 245378; 494^2 = (500-6)^2 = 250000 - 6000 + 36 = 244036
Diff = 245378 - 244036 = 1342

m = 504 / 1342 ≈ 0.3756

Intercept b = [Σy - mΣx]/n = [70 - 0.3756*494]/7
0.3756*494 ≈ let’s compute: 0.3756*500 = 187.8, minus 0.3756*6≈2.2536 → 187.8 - 2.2536 = 185.5464
So 70 - 185.5464 = -115.5464
Divide by 7: ≈ -16.5066

Equation: y ≈ 0.376x - 16.51

b) Slope ≈ 0.376 — means for every additional inch in height, shoe size increases by about 0.376 sizes.

c) Predict shoe size for height 105 inches:

y = 0.376*105 - 16.51 = 39.48 - 16.51 = 22.97 → about 23

That’s extremely large — unrealistic, but mathematically correct based on model.

d) What does slope represent? It represents how much shoe size changes per one-inch increase in height. In this case, roughly 0.38 shoe sizes per inch.

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Problem 3: Meal Cost vs Tip Left

Data:
Meal cost (x): 4.75, 6.84, 10.62, 10.42, 8.97
Tip left (y): 0.85, 1.20, 1.85, 1.80, 1.50

Find line of best fit, then predict tip for $15.50 meal.

First, check if proportional:

0.85 / 4.75 ≈ 0.1789
1.20 / 6.84 ≈ 0.1754
1.85 / 10.62 ≈ 0.1742
1.80 / 10.42 ≈ 0.1727
1.50 / 8.97 ≈ 0.1672

Very close to 17.5% — probably tip is 17.5% of meal cost.

Let’s verify with regression.

n=5
Σx = 4.75+6.84+10.62+10.42+8.97 = let’s add:
4.75+6.84=11.59; +10.62=22.21; +10.42=32.63; +8.97=41.60
Σy = 0.85+1.20+1.85+1.80+1.50 = 7.20
Σxy = 4.75*0.85 + 6.84*1.20 + 10.62*1.85 + 10.42*1.80 + 8.97*1.50

Compute each:

4.75*0.85 = 4.0375
6.84*1.20 = 8.208
10.62*1.85 = let’s compute: 10*1.85=18.5, 0.62*1.85≈1.147 → total ≈19.647
10.42*1.80 = 18.756
8.97*1.50 = 13.455
Sum xy: 4.0375 + 8.208 = 12.2455; +19.647 = 31.8925; +18.756 = 50.6485; +13.455 = 64.1035

Σx² = 4.75²=22.5625, 6.84²=46.7856, 10.62²=112.7844, 10.42²=108.5764, 8.97²=80.4609
Sum: 22.5625+46.7856=69.3481; +112.7844=182.1325; +108.5764=290.7089; +80.4609=371.1698

Slope m = [nΣxy - ΣxΣy] / [nΣx² - (Σx)^2]
= [5*64.1035 - 41.60*7.20] / [5*371.1698 - (41.60)^2]

Numerator: 5*64.1035 = 320.5175; 41.60*7.20 = 299.52; diff = 320.5175 - 299.52 = 20.9975

Denominator: 5*371.1698 = 1855.849; 41.60^2 = 1730.56; diff = 1855.849 - 1730.56 = 125.289

m = 20.9975 / 125.289 ≈ 0.1676

Intercept b = [Σy - mΣx]/n = [7.20 - 0.1676*41.60]/5
0.1676*41.60 ≈ 6.97216
7.20 - 6.97216 = 0.22784
/5 = 0.045568 ≈ 0.046

Equation: y ≈ 0.1676x + 0.046

But notice: when x=0, tip≈0.046 — almost zero. And slope ~0.1676 ≈ 16.76%, very close to 1/6 ≈ 16.67%.

Given the ratios earlier were all near 17.5%, but regression gives 16.76%. Perhaps due to rounding.

Alternatively, maybe the intended answer is simply 17.5% since all tips are approximately that.

Check average ratio: total tip / total cost = 7.20 / 41.60 ≈ 0.17307 → 17.3%

Close enough.

For prediction: if meal cost = $15.50

Using regression: y = 0.1676*15.50 + 0.046 ≈ 2.5978 + 0.046 ≈ 2.6438 → $2.64

Using 17.3%: 15.50 * 0.173 ≈ 2.6815 → $2.68

But let’s see what the problem expects. Since it says “use your calculator”, probably wants regression.

However, looking at the data again — perhaps it’s designed to be exactly 17.5%? Let’s test:

4.75 * 0.175 = 0.83125 ≈ 0.85? Close
6.84 * 0.175 = 1.197 ≈ 1.20 ✔️
10.62 * 0.175 = 1.8585 ≈ 1.85 ✔️
10.42 * 0.175 = 1.8235 ≈ 1.80? A bit off
8.97 * 0.175 = 1.56975 ≈ 1.50? Off by 0.07

Not perfect.

Regression is better.

So equation: y = 0.168x + 0.046 (rounded)

Predict for x=15.50:

y = 0.168*15.50 + 0.046 = 2.604 + 0.046 = 2.65

Or using more precise: 0.1676*15.5 = 2.5978 + 0.0456 = 2.6434 → $2.64

I think $2.64 is fine.

But let’s double-check calculations.

Earlier:

m = 20.9975 / 125.289 = let’s divide: 20.9975 ÷ 125.289 ≈ ?

125.289 × 0.1676 = 125.289 × 0.16 = 20.04624; 125.289 × 0.0076 = 0.9521964; total 20.9984364 — matches numerator 20.9975 closely.

So m=0.1676 is good.

b= (7.20 - 0.1676*41.60)/5

0.1676*41.60:

0.1676 * 40 = 6.704
0.1676 * 1.6 = 0.26816
Total 6.97216

7.20 - 6.97216 = 0.22784
/5 = 0.045568

So y = 0.1676x + 0.0456

At x=15.50:

0.1676 * 15.50 = 0.1676*15 + 0.1676*0.5 = 2.514 + 0.0838 = 2.5978
+ 0.0456 = 2.6434 → rounds to $2.64

Final Answer:

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Final Answer:
1b) y ≈ 7.86x + 62.52
1c) 109.68 (but note: likely capped at 100 in reality)
1d) No, because scores cannot exceed 100 and study time has limits.

2b) Slope ≈ 0.376 — shoe size increases by about 0.376 per inch of height.
2c) Approximately 23
2d) The slope represents the change in shoe size per one-inch increase in height.

3) Equation: y ≈ 0.168x + 0.046
For $15.50 meal, expected tip ≈ $2.64