Answers to Math Exercises & Math Problems: Angle Unit Conversion ... - Free Printable

It looks like you've uploaded a list of angle conversions involving degrees, minutes, and seconds (DMS), as well as decimal degrees. The task appears to be verifying or solving each conversion between decimal degrees and degrees-minutes-seconds (DMS) format.

Let’s go through each item one by one and check whether the given conversion is correct, explaining how to convert between formats.

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🔷 Key Concepts:

1. 1 degree = 60 minutes (′)
2. 1 minute = 60 seconds (″)
3. So:
- To convert decimal degrees → DMS:
- Degrees = integer part
- Minutes = (fractional part × 60) → take integer part
- Seconds = (remaining fractional part × 60)
- To convert DMS → decimal degrees:
- Decimal = degrees + (minutes / 60) + (seconds / 3600)

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We’ll now evaluate each equation.

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✔ a) $ 124^\circ = 2^\circ 04' $

✘ Incorrect

- Left side: 124°
- Right side: 2° 04′ = 2 + 4/60 ≈ 2.0667° ≠ 124°

Error: This is not equivalent.

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✔ b) $ 3^\circ 15' = 195' $

Let’s convert left side to minutes:
- 3° = 3 × 60 = 180′
- Plus 15′ → total = 195′

✔ Correct

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✔ c) $ 7.5^\circ = 450' $

Convert 7.5° to minutes:
- 7.5 × 60 = 450′

✔ Correct

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✔ d) $ 0.8^\circ = 0^\circ 48' $

Convert 0.8° to minutes:
- 0.8 × 60 = 48′

So 0.8° = 0°48′

✔ Correct

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✔ e) $ 324' = 5^\circ 24' $

Convert 324′ to degrees:
- 324 ÷ 60 = 5.4° → 5° + 0.4×60 = 24′ → 5°24′

✔ Correct

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✔ f) $ 17^\circ 12' = 1,032' $

Convert left side to minutes:
- 17° = 17 × 60 = 1020′
- Add 12′ → 1032′

✔ Correct

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✔ g) $ 1,230' = 20^\circ 30' $

Convert 1230′ to degrees:
- 1230 ÷ 60 = 20.5° → 20° + 0.5×60 = 30′ → 20°30′

✔ Correct

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✔ h) $ 6.25^\circ = 6^\circ 15' $

Break down:
- 6° remains
- 0.25° × 60 = 15′

So 6.25° = 6°15′

✔ Correct

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✔ i) $ 10.4^\circ = 624' $

Convert 10.4° to minutes:
- 10.4 × 60 = 624′

✔ Correct

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✔ j) $ 582' = 9^\circ 42' $

Convert 582′ to degrees:
- 582 ÷ 60 = 9.7° → 9° + 0.7×60 = 42′ → 9°42′

✔ Correct

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✔ k) $ 45.3^\circ = 2,718' $

Convert 45.3° to minutes:
- 45.3 × 60 = 2718′

✔ Correct

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✔ l) $ 16.05^\circ = 16^\circ 03' $

Break down:
- 16°
- 0.05° × 60 = 3′ → so 16°3′

✔ Correct

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✔ m) $ 52^\circ 27' = 3,147' $

Convert to minutes:
- 52° = 52 × 60 = 3120′
- Add 27′ → 3147′

✔ Correct

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✔ n) $ 1,000' = 16^\circ 40' $

Convert 1000′ to degrees:
- 1000 ÷ 60 = 16.666...° → 16° + 0.666...×60 = 40′ → 16°40′

✔ Correct

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✔ o) $ 11^\circ 59' = 719' $

Convert:
- 11° = 660′
- Add 59′ → 719′

✔ Correct

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✔ p) $ 81.6^\circ = 81^\circ 36' $

Break down:
- 0.6° × 60 = 36′ → yes

✔ Correct

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✔ q) $ 175.75^\circ = 10,545' $

Convert:
- 175.75 × 60 = ?
- 175 × 60 = 10,500
- 0.75 × 60 = 45
- Total = 10,545′

✔ Correct

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✔ r) $ 115.85^\circ = 115^\circ 51' $

Break down:
- 0.85° × 60 = 51′ → yes

✔ Correct

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✔ s) $ 87^\circ 09' = 5,229' $

Convert:
- 87° = 87 × 60 = 5220′
- Add 9′ → 5229′

✔ Correct

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✔ t) $ 521' = 8^\circ 41' $

Convert 521′ to degrees:
- 521 ÷ 60 = 8.6833...
- 8° + 0.6833×60 ≈ 41′ → 8°41′

✔ Correct

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✔ u) $ 274.65^\circ = 274^\circ 39' $

Break down:
- 0.65° × 60 = 39′ → yes

✔ Correct

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✔ v) $ 31.6^\circ = 1,896' $

Convert:
- 31.6 × 60 = 1896′

✔ Correct

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✔ w) $ 105^\circ 02' = 6,302' $

Convert:
- 105° = 105 × 60 = 6300′
- Add 2′ → 6302′

✔ Correct

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✔ x) $ 675.22^\circ = 40,513.2' $

Convert:
- 675.22 × 60 = ?
- 675 × 60 = 40,500
- 0.22 × 60 = 13.2
- Total = 40,513.2′

✔ Correct

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✔ y) $ 506.25' = 8^\circ 26' 15'' $

Convert 506.25′ to DMS:

First, convert to degrees:
- 506.25 ÷ 60 = 8.4375°
- Degrees: 8°
- Fractional part: 0.4375 × 60 = 26.25′ → 26′ and 0.25′ × 60 = 15″

So: 8°26′15″

✔ Correct

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✔ z) $ 22,500'' = 6.25^\circ $

Convert seconds to degrees:
- 22,500 ÷ 3600 = 6.25°

✔ Correct

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✔ Z) $ 19^\circ 19' 19'' = 69,559'' $

Convert to seconds:
- 19° = 19 × 3600 = 68,400″
- 19′ = 19 × 60 = 1,140″
- 19″ = 19″
- Total = 68,400 + 1,140 + 19 = 69,559″

✔ Correct

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✔ Final Summary:

All equations are correct except (a).

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✘ Only incorrect one:

> a) $ 124^\circ = 2^\circ 04' $ → False

This should be:

- $ 124^\circ = 124^\circ 00' $, or if converted to minutes: $ 124 \times 60 = 7440' $
- But $ 2^\circ 04' = 124' $, which is much smaller.

So clearly not equal.

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✔ Final Answer:

Only (a) is incorrect. All others are correct.

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If the task was to identify errors, then:

> Answer: (a) is false; all others are true.

Let me know if you want a corrected version of (a) or help with any specific conversion!