Ray's Arithmetic · Book 2 · 1877

Ray's New Intellectual Arithmetic

80 lessons of mental arithmetic, from adding in your head to fractions, ratio, percentage and interest, for children about 7 to 10. Every problem is solved aloud and explained, and each lesson has a guide.

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How to use it

  • Out loud, in the head. “Intellectual” means mental. Children answer aloud and then say why, following the model solutions printed in the book. Paper is only for jotting.
  • A little each day. About 20 minutes a day. Most lessons have 20 to 40 problems; spread them over two or three days.
  • Where to start. After Ray's New Primary Arithmetic, the book says you may begin at Lesson IV.
  • Answers. Each guide gives the answers that could be read clearly from the scan. Ray's own Key has them all, with worked solutions (see the book list).
  • Reasoning is the goal. Many later problems are puzzles. Let your child struggle a little, then model the first half of the explanation together.

Unit 1 Adding and subtracting in your head

Addition and subtraction to about 100, said aloud, with the reason for every answer. Lessons 1–11.

  1. 1Adding small numbers, and saying whyLesson I✓
  2. 2Adding drills and counting onLesson II✓
  3. 3The addition tableLesson III✓
  4. 4Adding to two-digit numbersLesson IV✓
  5. 5Adding several numbers, and counting by any numberLesson V✓
  6. 6SubtractionLesson VI✓
  7. 7The subtraction tableLesson VII✓
  8. 8Subtraction stories: how much more, how much gainedLesson VIII✓
  9. 9Review: adding and subtracting togetherLesson IX✓
  10. 10Chains with + and −Lesson X✓
  11. 11Mixed adding and subtracting storiesLesson XI✓

Unit 2 Multiplying and dividing

The tables to 12 × 12, then reasoning problems: from one thing to many, and from many to one. Lessons 12–19.

  1. 12Multiplying: the cost of severalLesson XII✓
  2. 13The multiplication table to 12 × 12Lesson XIII✓
  3. 14Multiplying in storiesLesson XIV✓
  4. 15Review: adding, subtracting and multiplyingLesson XV✓
  5. 16Dividing: how many can you buy?Lesson XVI✓
  6. 17The division tableLesson XVII✓
  7. 18Sharing, rates and the cost of oneLesson XVIII✓
  8. 19From one to many, and many to oneLesson XIX✓

Unit 3 What fractions are

Halves, thirds and fourths of real things, writing fractions, and changing them from one form to another. Lessons 20–29.

  1. 20Halves, thirds and other parts of a wholeLesson XX✓
  2. 21Writing fractionsLesson XXI✓
  3. 22A fraction of a numberLesson XXII✓
  4. 23What part of one number is another?Lesson XXIII✓
  5. 24Finding the whole from a partLesson XXIV✓
  6. 25Dividing with a fraction in the answerLesson XXV✓
  7. 26Mixed numbers into improper fractionsLesson XXVI✓
  8. 27Lowest termsLesson XXVII✓
  9. 28Changing a fraction to a new denominatorLesson XXVIII✓
  10. 29Common denominatorsLesson XXIX✓

Unit 4 Working with fractions

Adding, subtracting, multiplying and dividing fractions, all in the head, with story problems. Lessons 30–44.

  1. 30Adding fractionsLesson XXX✓
  2. 31Subtracting fractionsLesson XXXI✓
  3. 32Adding and subtracting together: what is left?Lesson XXXII✓
  4. 33Multiplying a fraction by a whole numberLesson XXXIII✓
  5. 34Rates with fractional answersLesson XXXIV✓
  6. 35Multiplying mixed numbersLesson XXXV✓
  7. 36Paying with something else (barter)Lesson XXXVI✓
  8. 37A fraction of a fractionLesson XXXVII✓
  9. 38Prices and amounts with fractionsLesson XXXVIII✓
  10. 39Finding the whole, with fractionsLesson XXXIX✓
  11. 40A fraction of a number is how many times another?Lesson XL✓
  12. 41Dividing by a fractionLesson XLI✓
  13. 42Dividing fractions by whole numbersLesson XLII✓
  14. 43Dividing by fractions in storiesLesson XLIII✓
  15. 44Shortcuts: parts of a dozen and of a dollarLesson XLIV✓

Unit 5 Review

Two long mixed reviews before the book turns to measures and harder reasoning. Lessons 45–46.

  1. 45Review: mixed problemsLesson XLV✓
  2. 46Review: the big mixed setLesson XLVI✓

Unit 6 Money, weights and measures

The tables of money, measure, weight, length and time, and problems that use them. Lessons 47–54.

  1. 47United States moneyLesson XLVII✓
  2. 48Dry measureLesson XLVIII✓
  3. 49Liquid measureLesson XLIX✓
  4. 50Avoirdupois weight (pounds and ounces)Lesson L✓
  5. 51Long measureLesson LI✓
  6. 52TimeLesson LII✓
  7. 53Problems with mixed unitsLesson LIII✓
  8. 54Business problems with whole numbersLesson LIV✓

Unit 7 Reasoning puzzles, ratio and proportion

Classic puzzles: parts of a number, ages, work done together, ratio, proportion and partnership. Lessons 55–63.

  1. 55Fractions of a price, and what is leftLesson LV✓
  2. 56Number puzzles: sums, differences and agesLesson LVI✓
  3. 57Dividing in fractional parts, and clock puzzlesLesson LVII✓
  4. 58A number and a part of itselfLesson LVIII✓
  5. 59Working togetherLesson LIX✓
  6. 60RatioLesson LX✓
  7. 61Ratio with three or more partsLesson LXI✓
  8. 62Proportion by reasoningLesson LXII✓
  9. 63PartnershipLesson LXIII✓

Unit 8 Percentage and interest

Per cent, gain and loss, discounts, commission, insurance, and simple interest every way round. Lessons 64–79.

  1. 64Per cent: what it meansLesson LXIV✓
  2. 65Finding a per cent of a numberLesson LXV✓
  3. 66Fractions as per centsLesson LXVI✓
  4. 67What per cent? Gain and lossLesson LXVII✓
  5. 68Finding the cost from the selling priceLesson LXVIII✓
  6. 69Discounts: “per cent off”Lesson LXIX✓
  7. 70CommissionLesson LXX✓
  8. 71InsuranceLesson LXXI✓
  9. 72Simple interest for whole yearsLesson LXXII✓
  10. 73Interest for months and daysLesson LXXIII✓
  11. 74Finding the principal from the interestLesson LXXIV✓
  12. 75Finding the principal from the amountLesson LXXV✓
  13. 76Finding the timeLesson LXXVI✓
  14. 77Finding the rateLesson LXXVII✓
  15. 78Present worth and discountLesson LXXVIII✓
  16. 79Interest puzzlesLesson LXXIX✓

Unit 9 Miscellaneous

The book ends with sixty-one mixed problems. Spread them over several weeks. Lessons 80–80.

  1. 80Miscellaneous problemsLesson LXXX✓

The book itself

Ray's New Intellectual Arithmetic

1877 · Mental arithmetic, about ages 7 to 10

Arithmetic done in the head and explained aloud: the four operations with larger numbers, fractions, and many story problems. Ray's “intellectual” means mental.

🧑‍🏫 Teaching guides ⬇ Download PDF (7.9 MB) Internet Archive ↗

Scanned from a University of California copy.

The answers

Key to Ray's New Arithmetic: Intellectual and Practical

later printing (1907 scan) · For the parent

The answers and worked solutions for the Intellectual and Practical books.

⬇ Download PDF (18 MB) Internet Archive ↗

Scanned printing dated 1907.

Lesson numbers match the book's Roman numerals: Lesson 3 is the book's Lesson III. The guides are written for this site; the book is in the public domain.

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